{"id":16765,"date":"2022-04-02T13:08:40","date_gmt":"2022-04-02T07:38:40","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=16765"},"modified":"2022-04-07T10:58:31","modified_gmt":"2022-04-07T05:28:31","slug":"big-ideas-math-answers-grade-6-chapter-1","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/big-ideas-math-answers-grade-6-chapter-1\/","title":{"rendered":"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors"},"content":{"rendered":"<p><strong>Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors:<\/strong> Before knowing the numerical expressions, Grade 6 students must be familiar with composite numbers, factor pairs, and prime numbers. Refer to the concept of vocabulary terms, subtracting and adding the fractions, mixed fractions in the below sections.<\/p>\n<p>Know the various terms like what is exponent, perfect square, power, composite numbers, prime numbers with the help of <a href=\"https:\/\/bigideasmathanswers.com\/big-ideas-math-answers-grade-6\/\">Big Ideas Math Answers Grade 6<\/a> Chapter 1 Numerical Expressions and Factors. You can also get all the answers to the questions which are available in the Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors pdf format. Scroll to the below sections to check example problems, answer PDFs, etc.<\/p>\n<h2>Big Ideas Math Book 6th Grade Answer Key Chapter 1 Numerical Expressions and Factors<\/h2>\n<p>BIM 6th Grade Chapter 1 Numerical Expression and Factors Answer key helps you in easy and quick learning. <strong>Download Big Ideas Math Book 6th Grade Answer Key Chapter 1 Numerical Expressions and Factors<\/strong> pdf to kickstart your preparation. Get the solutions for all the questions in this article. There are various topics included in numerical expressions like Least Common Multiple, Prime Factorisation, Powers and Exponents, Order of Operations, Greatest Common Factor, Exponents and Powers. Click on the below links and prepare for the exam as per the topics.<\/p>\n<p><strong>Performance Task<\/strong><\/p>\n<ul>\n<li><a href=\"#Numerical_Expressions_and_Factors_Steam_Video\/_Performance_Task\">Numerical Expressions and Factors Steam Video\/Performance Task<\/a><\/li>\n<li><a href=\"#Numerical_Expressions_and_Factors_Getting_Ready_for_Chapter_1\">Numerical Expressions and Factors Getting Ready for Chapter 1<\/a><\/li>\n<\/ul>\n<p><strong>Lesson 1: Powers and Exponents<\/strong><\/p>\n<ul>\n<li><a href=\"#Lesson_1.1_Powers_and_Exponents\">Lesson 1.1 Powers and Exponents<\/a><\/li>\n<li><a href=\"#Powers_and_Exponents_Practice_1.1\">Powers and Exponents Practice 1.1<\/a><\/li>\n<\/ul>\n<p><strong>Lesson 2: Order of Operations<\/strong><\/p>\n<ul>\n<li><a href=\"#Lesson_1.2_Order_of_Operations\">Lesson 1.2 Order of Operations<\/a><\/li>\n<li><a href=\"#Order_of_Operations_Practice_1.2\">Order of Operations Practice 1.2<\/a><\/li>\n<\/ul>\n<p><strong>Lesson 3: Prime Factorization<\/strong><\/p>\n<ul>\n<li><a href=\"#Lesson_1.3_Prime_Factorization\">Lesson 1.3 Prime Factorization<\/a><\/li>\n<li><a href=\"#Prime_Factorization_Practice_1.3\">Prime Factorization Practice 1.3<\/a><\/li>\n<\/ul>\n<p><strong>Lesson 4: Greatest Common Factor<\/strong><\/p>\n<ul>\n<li><a href=\"#Lesson_1.4_Greatest_Common_Factor\">Lesson 1.4 Greatest Common Factor<\/a><\/li>\n<li><a href=\"#Greatest_Common_Factor_Practice_1.4\">Greatest Common Factor Practice 1.4<\/a><\/li>\n<\/ul>\n<p><strong>Lesson 5: Least Common Multiple<\/strong><\/p>\n<ul>\n<li><a href=\"#Lesson_1.5_Least_Common_Multiple\">Lesson 1.5 Least Common Multiple<\/a><\/li>\n<li><a href=\"#Least_Common_Multiple_Practice_1.5\">Least Common Multiple Practice 1.5<\/a><\/li>\n<\/ul>\n<p><strong>Chapter: 1 &#8211; Numerical Expressions and Factors<\/strong><\/p>\n<ul>\n<li><a href=\"#Numerical_Expressions_and_Factors_Connecting_Concepts\">Numerical Expressions and Factors Connecting Concepts<\/a><\/li>\n<li><a href=\"#Numerical_Expressions_and_Factors_Chapter_Review\">Numerical Expressions and Factors Chapter Review<\/a><\/li>\n<li><a href=\"#Numerical_Expressions_and_Factors_Practice_Test\">Numerical Expressions and Factors Practice Test<\/a><\/li>\n<li><a href=\"#Numerical_Expressions_and_Factors_Cumulative_Practice\">Numerical Expressions and Factors Cumulative Practice<\/a><\/li>\n<\/ul>\n<h3><a id=\"Numerical_Expressions_and_Factors_Steam_Video\/_Performance_Task\"><\/a>Numerical Expressions and Factors Steam Video\/Performance Task<\/h3>\n<p><strong>Filling Pi\u00f1atas<\/strong><\/p>\n<p>Common factors can be used to make identical groups of objects. Can you think of any situations in which you would want to separate objects into equal groups? Are there any common factors that may be more useful than others? Can you think of any other ways to use common factors?<br \/>\nwatch the STEAM Video \u201cFilling Pi\u00f1atas.\u201d Then answer the following questions. The table below shows the numbers of party favors that Alex and Enid use to make pi\u00f1atas.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16951\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-1.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 1\" width=\"242\" height=\"250\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16950\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-1.1.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 1.1\" width=\"426\" height=\"83\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-1.1.png 426w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-1.1-300x58.png 300w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><\/p>\n<p>Question 1.<br \/>\nWhen \ufb01nding the number of identical pi\u00f1atas that can be made, why is it helpful for Alex and Enid to list the factors of each number given in the table?<\/p>\n<p>Answer: By using the list of the factors of all the numbers Alex and Enid can make identical groups of the objects.<\/p>\n<p>Question 2.<br \/>\nYou want to create 6 identical pi\u00f1atas. How can you change the numbers of party favors in the table to make this happen? Can you do this without changing the total number of party favors?<\/p>\n<p>Answer: You can change the number of party favors to create 6 identical pinatas.<br \/>\nThere are 100 Mints. So divide it into two identical groups.<br \/>\nChange the number of mints to 50. And add 50 to new identical pinatas.<\/p>\n<p><strong>Performance Task<\/strong><\/p>\n<p><strong>Setting the Table<\/strong><\/p>\n<p>After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be asked to plan a fundraising event with the items below.<br \/>\n<strong>72 chairs<\/strong><br \/>\n<strong>48 balloons<\/strong><br \/>\n<strong>24 flowers<\/strong><br \/>\n<strong>36 candles<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16952\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-2.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 2\" width=\"290\" height=\"363\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-2.png 290w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-2-240x300.png 240w\" sizes=\"(max-width: 290px) 100vw, 290px\" \/><br \/>\nYou will \ufb01nd the greatest number of identical tables that can be prepared, and what will be in each centerpiece. When making arrangements for a party, should a party planner always use the greatest number of identical tables possible? Explain why or why not.<\/p>\n<p>Answer:<br \/>\n72 chairs = 2 \u00d7 36<br \/>\n= 2 \u00d7 2 \u00d7 18<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 9<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 3 \u00d7 3<br \/>\n2, 2, 2, 3, 3<br \/>\nTherefore, <strong>1, 2, 3, 4, 6, <\/strong>8, 9, <strong>12<\/strong>, 18, 24, 36, and 72 are the factors of 72.<br \/>\n48 balloons = 2 \u00d7 24<br \/>\n= 2 \u00d7 2 \u00d7 12<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 6<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 3<br \/>\nThe positive Integer factors of 48 are <strong>1, 2, 3, 4, 6, <\/strong>8, <strong>12<\/strong>, 16, 24 and 48.<br \/>\nThe factors of number 24 are <strong>1, 2, 3, 4, 6, <\/strong>8, <strong>12, <\/strong>24.<br \/>\nThe factors of number 36 are <strong>1, 2, 3, 4, 6<\/strong>, 9, <strong>12<\/strong>, 18, and 36<br \/>\nThe greatest number of identical tables possible are 1, 2, 3, 4, 6, 12.<\/p>\n<h3><a id=\"Numerical_Expressions_and_Factors_Getting_Ready_for_Chapter_1\"><\/a>Numerical Expressions and Factors Getting Ready for Chapter 1<\/h3>\n<p><strong>Chapter Exploration<\/strong><\/p>\n<p><strong>Work with a partner. In Exercises 1\u20133, use the table.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16953\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 3\" width=\"623\" height=\"451\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3.png 623w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3-300x217.png 300w\" sizes=\"(max-width: 623px) 100vw, 623px\" \/><\/p>\n<ol>\n<li>Cross out the multiples of 2 that are greater than 2. Do the same for 3, 5, and 7.<\/li>\n<li>The numbers that are not crossed out are called prime numbers. The numbers that are crossed out are called composite numbers. In your own words, describe the characteristics of prime numbers and composite numbers.<\/li>\n<li><strong>MODELING REAL LIFE<\/strong> Work with a partner. Cicadas are insects that live underground and emerge from the ground after x or x + 4 years. Is it possible that both x and x +4 are prime? Give some examples.<\/li>\n<\/ol>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-22313\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3-300x217.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3\" width=\"300\" height=\"217\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3-300x217.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-3.jpg 623w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nThe numbers that are not crossed are 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71, 73, 79, 83, 89, 97.<br \/>\nThese are not multiples of any numbers. So, the above numbers are the prime numbers.<\/p>\n<p><strong>Vocabulary<\/strong><\/p>\n<p>The following vocabulary terms are de\ufb01ned in this chapter. Think about what each term might mean and record your thoughts.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16954\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-4.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 4\" width=\"585\" height=\"76\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-4.png 585w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-4-300x39.png 300w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>Answer:<br \/>\ni. First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.<br \/>\nii. &#8220;Factors&#8221; are numbers we can multiply together to get another number. When we find the factors of two or more numbers, and then find some factors are the same, then they are the &#8220;common factors&#8221;.<br \/>\niii. A common multiple is a whole number that is a shared multiple of each set of numbers. The multiples that are common to two or more numbers are called the common multiples of those numbers. The smallest positive number is a multiple of two or more numbers.<\/p>\n<h3><a id=\"Lesson_1.1_Powers_and_Exponents\"><\/a>Lesson 1.1 Powers and Exponents<\/h3>\n<p><strong>Exploration 1<\/strong><br \/>\n<strong>Writing Expressions Using Exponents<\/strong><br \/>\n<strong>Work with a partner. Copy and complete the table.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16955\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 5\" width=\"707\" height=\"327\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5.png 707w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5-300x139.png 300w\" sizes=\"(max-width: 707px) 100vw, 707px\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-22035\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5-300x139.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5\" width=\"300\" height=\"139\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5-300x139.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-5.jpg 707w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\ni. In your own words, describe what the two numbers in the expression 3<sup>5<\/sup> mean.<\/p>\n<p>Answer: 3<sup>5<\/sup> means the number 3 repeats 5 times.<br \/>\n3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3 = 243<\/p>\n<p><strong>EXPLORATION 2<\/strong><br \/>\n<strong>Using a Calculator to Find a Pattern<\/strong><\/p>\n<p>Work with a partner. Copy the diagram. Use a calculator to \ufb01nd each value. Write one digit of the value in each box. Describe the pattern in the digits of the values.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16956\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 6\" width=\"445\" height=\"195\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6.png 445w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6-300x131.png 300w\" sizes=\"(max-width: 445px) 100vw, 445px\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-22036\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6-300x131.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6\" width=\"300\" height=\"131\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6-300x131.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-6.jpg 445w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><strong>1.1 Lesson<\/strong><\/p>\n<p>A power is a product of repeated factors. The base of a power is the repeated factor. The exponent of a power indicates the number of times the base is used as a factor.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16957\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-7.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 7\" width=\"346\" height=\"103\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-7.png 346w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-7-300x89.png 300w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16958\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-8.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 8\" width=\"645\" height=\"211\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-8.png 645w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-8-300x98.png 300w\" sizes=\"(max-width: 645px) 100vw, 645px\" \/><\/p>\n<p><strong>Try It<\/strong><br \/>\nWrite the product as a power.<br \/>\nQuestion 1.<br \/>\n2 \u00d7 2 \u00d7 2<\/p>\n<p>Answer: 2\u00b3\u00a0 = 8<br \/>\nTwo cubed or three to the two. Here 2 is repeated three times.<\/p>\n<p>Question 2.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16959\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-13.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 13\" width=\"130\" height=\"32\" \/><\/p>\n<p>Answer: 46656 = 6<sup>6<br \/>\n<\/sup><\/p>\n<p>Six to the six. Here 6 is repeated six times.<\/p>\n<p>Question 3.<br \/>\n15 \u00d7 15 \u00d7 15 \u00d7 15<\/p>\n<p>Answer: 56025 = 15<sup>4<\/sup><\/p>\n<p>15 to the power 4. Here 15 is repeated four times.<\/p>\n<p>Question 4.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16960\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-14.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 14\" width=\"221\" height=\"28\" \/><\/p>\n<p>Answer: 1280000000 = 20<sup>7<\/sup><\/p>\n<p>20 to the power 7. Here 20 is repeated seven times.<\/p>\n<p><strong>Try It<\/strong><br \/>\nFind the value of the power.<\/p>\n<p>Question 5.<br \/>\n6<sup>3<\/sup><\/p>\n<p>Answer: 6 \u00d7 6 \u00d7 6 = 216<br \/>\nThe value of the power 6<sup>3 <\/sup>is 216.<\/p>\n<p>Question 6.<br \/>\n9<sup>2<\/sup><\/p>\n<p>Answer: 9 \u00d7 9 = 81<br \/>\nThe value of the power 9<sup>2<\/sup> is 81<\/p>\n<p>Question 7.<br \/>\n3<sup>4<\/sup><\/p>\n<p>Answer: 3 \u00d7 3 \u00d7 3 \u00d7 3<br \/>\nThe value of the power 3<sup>4 <\/sup>is 81.<\/p>\n<p>Question 8.<br \/>\n18<sup>2<\/sup><\/p>\n<p>Answer: 18 \u00d7 18<br \/>\nThe value of the power 18<sup>2<\/sup> is 324.<\/p>\n<p><strong>Try It<\/strong><br \/>\nDetermine whether the number is a perfect square.<\/p>\n<p>Question 9.<br \/>\n25<\/p>\n<p>Answer: 5\u00b2<br \/>\nYes, 25 is the perfect square.<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p>Question 10.<br \/>\n2<\/p>\n<p>Answer: 2 is not a perfect square. 2 cannot be expressed as the square of a number from the same number system.<\/p>\n<p>Question 11.<br \/>\n99<\/p>\n<p>Answer: 99 is not a perfect square. 99 cannot be expressed as the square of a number from the same number system.<\/p>\n<p>Question 12.<br \/>\n36<\/p>\n<p>Answer: 6\u00b2<br \/>\n36 is a perfect square<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p><strong>Self-Assessment for Concepts &amp; Skills<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<br \/>\n<strong>FINDING VALUES OF POWERS<\/strong><br \/>\n<strong>Find the value of the power.<\/strong><\/p>\n<p>Question 13.<br \/>\n8<sup>2<\/sup><\/p>\n<p>Answer: 8 \u00d7 8 = 64<br \/>\nThe value of the power 8<sup>2 <\/sup>is 64.<\/p>\n<p>Question 14.<br \/>\n3<sup>5<\/sup><\/p>\n<p>Answer: 3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3 = 243<br \/>\nThe value of the power 3<sup>5 <\/sup>is 243.<\/p>\n<p>Question 15.<br \/>\n11<sup>3<\/sup><\/p>\n<p>Answer: 11 \u00d7 11 \u00d7 11 = 1331<br \/>\nThe value of the power 11<sup>3 <\/sup>is 1331.<\/p>\n<p>Question 16.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nHow are exponents and powers different?<\/p>\n<p>Answer:<br \/>\nAn expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.<\/p>\n<p>Question 17.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nIs 10 a perfect square? Is 100 a perfect square? Explain.<\/p>\n<p>Answer: 10 is not a perfect square.<br \/>\nA perfect square is a number that is generated by multiplying two equal integers by each other.<br \/>\n100 is a perfect square. Because 10 \u00d7 10 = 100.<\/p>\n<p>Question 18.<br \/>\n<strong>WHICH ONE DOESN\u2019T BELONG?<\/strong><br \/>\nWhich one does not belong with the other three? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16961\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-15.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 15\" width=\"341\" height=\"97\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-15.png 341w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-15-300x85.png 300w\" sizes=\"(max-width: 341px) 100vw, 341px\" \/><\/p>\n<p>Answer:<br \/>\n2<sup>4<\/sup> = 2 \u00d7 2 \u00d7 2 \u00d7 2 = 16<br \/>\n3<sup>2<\/sup> = 3 \u00d7 3 = 9<br \/>\n3 + 3 + 3 + 3 = 3 \u00d7 4<br \/>\n5.5.5 = 125<br \/>\nThe 3rd option does not belong to the other three expressions.<\/p>\n<p><strong>Self-Assessment for Problem Solving<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p>Question 19.<br \/>\nA square solar panel has an area of 16 square feet. Write the area as a power. Then \ufb01nd the side lengths of the panel.<\/p>\n<p>Answer: 4 feet<\/p>\n<p>Explanation:<br \/>\nGiven that,<br \/>\nA square solar panel has an area of 16 square feet.<br \/>\nA = s \u00d7 s<br \/>\n16 = s\u00b2<br \/>\n4\u00b2 = s\u00b2<br \/>\ns = 4<br \/>\nThus the side length of the panel is 4 feet.<\/p>\n<p>Question 20.<br \/>\nThe four-square court shown is a square made up of four identical smaller squares. What is the area of the court?<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-16962\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-16.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 16\" width=\"170\" height=\"159\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nThe four-square court shown is a square made up of four identical smaller squares.<br \/>\nThe side of each square is 6 feet.<br \/>\n6 + 6 = 12 feet<br \/>\nThe area of the court is 12 ft \u00d7 12 ft<br \/>\nA = 144 square feet<br \/>\nThus the area of the court is 144 square feet.<\/p>\n<p>Question 21.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nEach face of a number cube is a square with a side length of 16 millimeters. What is the total area of all of the faces of the number cube?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nEach face of a number cube is a square with a side length of 16 millimeters.<br \/>\nArea of the cube = 6 a\u00b2<br \/>\nA = 6 \u00d7 16 \u00d7 16<br \/>\nA = 1536 sq. mm<\/p>\n<h3><a id=\"Powers_and_Exponents_Practice_1.1\"><\/a>Powers and Exponents<span id=\"P20_52\" class=\"s1 f60 fs30 l0 t12 o0 tr20\"> <\/span>Practice 1.1<\/h3>\n<p><strong>Review &amp; Refresh<\/strong><\/p>\n<p>Multiply.<\/p>\n<p>Question 1.<br \/>\n150 \u00d7 2<\/p>\n<p>Answer: 300<\/p>\n<p>Explanation:<br \/>\nMultiply the two numbers 150 and 2.<br \/>\nFirst multiply 2 with ones place 2 \u00d7 0 = 0<br \/>\nNext multiply with tens place 2 \u00d7 50 = 100<br \/>\nNext multiply with hundreds place 2 \u00d7 100 = 200<br \/>\n200 + 100 = 300<\/p>\n<p>Question 2.<br \/>\n175 \u00d7 8<\/p>\n<p>Answer: 1400<\/p>\n<p>Explanation:<br \/>\nMultiply the two numbers 175 and 8.<br \/>\nFirst, multiply 2 with ones place 8 \u00d7 5 = 40<br \/>\nNext multiply with tens place 8 \u00d7 70= 560<br \/>\nNext multiply with hundreds place 8 \u00d7 100 = 800<br \/>\n800 + 560 + 40 = 1400<\/p>\n<p>Question 3.<br \/>\n123 \u00d7 3<\/p>\n<p>Answer: 369<\/p>\n<p>Explanation:<br \/>\nMultiply the two numbers 123 and 3.<br \/>\nFirst multiply 2 with ones place 3 \u00d7 3 = 9<br \/>\nNext multiply with tens place 3 \u00d7 20 = 60<br \/>\nNext multiply with hundreds place 3 \u00d7100 = 300<br \/>\n300 + 60 + 9 = 369<\/p>\n<p>Question 4.<br \/>\n151 \u00d7 9<\/p>\n<p>Answer: 1359<\/p>\n<p>Explanation:<br \/>\nMultiply the two numbers 151 and 9.<br \/>\nFirst multiply 2 with ones place 9 \u00d7 1 = 9<br \/>\nNext multiply with tens place 9 \u00d7 50 = 450<br \/>\nNext multiply with hundreds place 9 \u00d7 100 = 900<br \/>\n900 + 450 + 9 = 1359<\/p>\n<p>Write the sentence as a numerical expression.<\/p>\n<p>Question 5.<br \/>\nAdd 5 and 8, then multiply by 4.<\/p>\n<p>Answer: The numerical expression for the above sentence is 5 + 8 \u00d7 4<\/p>\n<p>Question 6.<br \/>\nSubtract 7 from 11, then divide by 2.<\/p>\n<p>Answer: The numerical expression for the above sentence is 11 &#8211; 7 \u00f7 2<br \/>\nRound the number to the indicated place value.<\/p>\n<p>Question 7.<br \/>\n4.03785 to the tenths<\/p>\n<p>Answer: The number 4.03785 nearest to the tenths is 4.0<\/p>\n<p>Question 8.<br \/>\n12.89503 to the hundredths<\/p>\n<p>Answer: The number 12.89503 nearest to the hundredths is 12.90<\/p>\n<p>Complete the sentence.<\/p>\n<p>Question 9.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16963\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-17.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 17\" width=\"122\" height=\"46\" \/><\/p>\n<p>Answer: 3<\/p>\n<p>Explanation:<br \/>\n(1\/10) \u00d7 30 = 30\/10 = 3<br \/>\nThe product of 1\/10 and 30 is 3.<\/p>\n<p>Question 10.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16964\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-18.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 18\" width=\"115\" height=\"50\" \/><\/p>\n<p>Answer: 20<\/p>\n<p>Explanation:<br \/>\n(4\/5) \u00d7 25 = 4 \u00d7 5 = 20<br \/>\nThe product of 4\/5 and 25 is 20.<\/p>\n<p><strong>Concepts, Skills, &amp; Problem Solving<\/strong><\/p>\n<p><strong>WRITING EXPRESSIONS USING EXPONENTS<\/strong><br \/>\n<strong>Copy and complete the table.<\/strong> (See Exploration 1, p. 3.)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16965\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 19\" width=\"430\" height=\"171\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19.png 430w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19-300x119.png 300w\" sizes=\"(max-width: 430px) 100vw, 430px\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-22375\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19-300x119.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19\" width=\"300\" height=\"119\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19-300x119.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-19.jpg 430w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>WRITING EXPRESSIONS AS POWERS<\/strong><br \/>\n<strong>Write the product as a power.<\/strong><\/p>\n<p>Question 15.<br \/>\n9 \u00d7 9<\/p>\n<p>Answer: The exponential form of the given expression is 9\u00b2<\/p>\n<p>Question 16.<br \/>\n13 \u00d7 13<\/p>\n<p>Answer: The exponential form of the given expression is 13\u00b2<\/p>\n<p>Question 17.<br \/>\n15 \u00d7 15 \u00d7 15<\/p>\n<p>Answer: The exponential form of the given expression is 15\u00b3<\/p>\n<p>Question 18.<br \/>\n2.2.2.2.2<\/p>\n<p>Answer: The exponential form of the given expression is 2<sup>5<\/sup><\/p>\n<p>Question 19.<br \/>\n14 \u00d7 14 \u00d7 14<\/p>\n<p>Answer: The exponential form of the given expression is 14\u00b3<\/p>\n<p>Question 20.<br \/>\n8.8.8.8<\/p>\n<p>Answer: The exponential form of the given expression is 8<sup>4<\/sup><\/p>\n<p>Question 21.<br \/>\n11 \u00d7 11 \u00d7 11 \u00d7 11 \u00d7 11<\/p>\n<p>Answer: The exponential form of the given expression is 11<sup>5<\/sup><\/p>\n<p>Question 22.<br \/>\n7.7.7.7.7.7<\/p>\n<p>Answer: The exponential form of the given expression is 7<sup>6<\/sup><\/p>\n<p>Question 23.<br \/>\n16.16.16.16<\/p>\n<p>Answer: The exponential form of the given expression is 16<sup>4<\/sup><\/p>\n<p>Question 24.<br \/>\n43 \u00d7 43 \u00d7 43 \u00d7 43 \u00d7 43<\/p>\n<p>Answer: The exponential form of the given expression is 43<sup>5<\/sup><\/p>\n<p>Question 25.<br \/>\n167 \u00d7 167 \u00d7 167<\/p>\n<p>Answer: The exponential form of the given expression is 167\u00b3<\/p>\n<p>Question 26.<br \/>\n245.245.245.245<\/p>\n<p>Answer: The exponential form of the given expression is 245<sup>4<\/sup><\/p>\n<p><strong>FINDING VALUES OF POWERS<\/strong><br \/>\n<strong>Find the value of the power.<\/strong><\/p>\n<p>Question 27.<br \/>\n5<sup>2<\/sup><\/p>\n<p>Answer: The value of the powers 5<sup>2 <\/sup>is 5 \u00d7 5 = 25<\/p>\n<p>Question 28.<br \/>\n4<sup>3<\/sup><\/p>\n<p>Answer: The value of the powers 4<sup>3<\/sup> is 4 \u00d7 4 \u00d7 4 = 64<\/p>\n<p>Question 29.<br \/>\n6<sup>2<\/sup><\/p>\n<p>Answer: The value of the powers 6<sup>2<\/sup> is 6 \u00d7 6 = 36<\/p>\n<p>Question 30.<br \/>\n1<sup>7<\/sup><\/p>\n<p>Answer: The value of the powers 1<sup>7<\/sup> is 1 \u00d7 1 \u00d7 1 \u00d7 1 \u00d7 1 \u00d7 1 \u00d7 1 = 1<\/p>\n<p>Question 31.<br \/>\n0<sup>3<\/sup><\/p>\n<p>Answer: The value of the powers 0<sup>3<\/sup> is 0 \u00d7 0 \u00d7 0 = 0<\/p>\n<p>Question 32.<br \/>\n8<sup>4<\/sup><\/p>\n<p>Answer: The value of the powers 8<sup>4<\/sup> is 8 \u00d7 8 \u00d7 8 \u00d7 8 = 4096<\/p>\n<p>Question 33.<br \/>\n2<sup>4<\/sup><\/p>\n<p>Answer: The value of the powers 2<sup>4<\/sup> is 2 \u00d7 2 \u00d7 2 \u00d7 2 = 64<\/p>\n<p>Question 34.<br \/>\n12<sup>2<\/sup><\/p>\n<p>Answer: The value of the powers 12<sup>2<\/sup> is 12 \u00d7 12 = 144<\/p>\n<p>Question 35.<br \/>\n7<sup>3<\/sup><\/p>\n<p>Answer: The value of the powers 7<sup>3 <\/sup>is 7 \u00d7 7 \u00d7 7 = 343<\/p>\n<p>Question 36.<br \/>\n5<sup>4<\/sup><\/p>\n<p>Answer: The value of the powers 5<sup>4 <\/sup>is 5 \u00d7 5 \u00d7 5 \u00d7 5 = 625<\/p>\n<p>Question 37.<br \/>\n2<sup>5<\/sup><\/p>\n<p>Answer: The value of the powers 2<sup>5<\/sup> is 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 = 32<\/p>\n<p>Question 38.<br \/>\n14<sup>2<\/sup><\/p>\n<p>Answer: The value of the powers 14<sup>2<\/sup> is 14 \u00d7 14 = 196<\/p>\n<p><strong>USING TOOLS<\/strong><br \/>\n<strong>Use a calculator to \ufb01nd the value of the power.<\/strong><\/p>\n<p>Question 39.<br \/>\n7<sup>6<\/sup><\/p>\n<p>Answer: 7 \u00d7 7 \u00d7 7 \u00d7 7 \u00d7 7 \u00d7 7 = 117649<\/p>\n<p>Question 40.<br \/>\n4<sup>8<\/sup><\/p>\n<p>Answer: 4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4 = 256<\/p>\n<p>Question 41.<br \/>\n12<sup>4<\/sup><\/p>\n<p>Answer: 12 \u00d7 12 \u00d7 12 \u00d7 12 = 20736<\/p>\n<p>Question 42.<br \/>\n17<sup>5<\/sup><\/p>\n<p>Answer: 17 \u00d7 17 \u00d7 17 \u00d7 17 \u00d7 17 = 1419857<\/p>\n<p>Question 43.<br \/>\n<strong>YOU BE THE TEACHER<\/strong><br \/>\nYour friend \ufb01nds the value of 8<sup>3<\/sup>. Is your friend correct? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16966\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-20.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 20\" width=\"203\" height=\"69\" \/><\/p>\n<p>Answer: Your friend is incorrect<br \/>\n8<sup>3<\/sup> is nothing but 8 repeats 3 times.<br \/>\n8<sup>3<\/sup> = 8 \u00d7 8 \u00d7 8 = 512<\/p>\n<p><strong>IDENTIFYING PERFECT SQUARES<\/strong><br \/>\n<strong>Determine whether the number is a perfect square.<\/strong><\/p>\n<p>Question 44.<br \/>\n8<\/p>\n<p>Answer: 8 is not the perfect square. 8 cannot be expressed as the square of a number from the same number system.<\/p>\n<p>Question 45.<br \/>\n4<\/p>\n<p>Answer: 4 is a perfect square.<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p>Question 46.<br \/>\n81<\/p>\n<p>Answer: 81 perfect square<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p>Question 47.<br \/>\n44<\/p>\n<p>Answer: 44 is not the perfect square. 44 cannot be expressed as the square of a number from the same number system<\/p>\n<p>Question 48.<br \/>\n49<\/p>\n<p>Answer: 49 is a perfect square<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p>Question 49.<br \/>\n125<\/p>\n<p>Answer: 125 is not the perfect square. 125 cannot be expressed as the square of a number from the same number system<\/p>\n<p>Question 50.<br \/>\n150<\/p>\n<p>Answer: 150 is not the perfect square. 150 cannot be expressed as the square of a number from the same number system<\/p>\n<p>Question 51.<br \/>\n144<\/p>\n<p>Answer: 144 is\u00a0the perfect square<br \/>\nA perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.<\/p>\n<p>Question 52.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nOn each square centimeter of a person\u2019s skin, there are about 39<sup>2<\/sup> bacteria. How many bacteria does this expression represent?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nOn each square centimeter of a person\u2019s skin, there are about 39<sup>2<\/sup> bacteria.<br \/>\n39<sup>2<\/sup> \u00a0= 39 \u00d7 39 = 1521 centimeters<br \/>\nThus the bacteria represents 1521 centimeters.<\/p>\n<p>Question 53.<br \/>\n<strong>REPEATED REASONING<\/strong><br \/>\nThe smallest \ufb01gurine in a gift shop is 2 inches tall. The height of each \ufb01gurine is twice the height of the previous \ufb01gurine. What is the height of the tallest \ufb01gurine?<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-16967\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-21.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 21\" width=\"423\" height=\"346\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-21.png 423w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-21-300x245.png 300w\" sizes=\"(max-width: 423px) 100vw, 423px\" \/><\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nThe smallest \ufb01gurine in a gift shop is 2 inches tall. The height of each \ufb01gurine is twice the height of the previous \ufb01gurine.<br \/>\nThe second \ufb01gurine is twice that of the first \ufb01gurine = 2 \u00d7 2 = 4 inches<br \/>\nThe third \ufb01gurine is twice that of the second \ufb01gurine = 4 \u00d7 4 = 16 inches<br \/>\nThe fourth \ufb01gurine is twice that of the third \ufb01gurine = 16 \u00d7 16 = 256 inches<br \/>\nThus the height of the tallest \ufb01gurine is 256 inches.<\/p>\n<p>Question 54.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nA square painting measures 2 meters on each side. What is the area of the painting in square centimeters?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nA square painting measures 2 meters on each side.<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 2 m \u00d7 2 m = 4 sq. meters<br \/>\nThus the area of the painting in square centimeters is 4.<\/p>\n<p>Question 55.<br \/>\n<strong>NUMBER SENSE<\/strong><br \/>\nWrite three powers that have values greater than 120 and less than 130.<\/p>\n<p>Answer:<br \/>\n11\u00b2 = 11(11) = 121; this is between 120 and 130.<br \/>\n5\u00b3 = 5(5)(5) = 25(5) = 125; this is between 120 and 130.<br \/>\n2\u2077 = 2(2)(2)(2)(2)(2)(2) = 4(2)(2)(2)(2)(2) = 8(2)(2)(2)(2) = 16(2)(2)(2) = 32(2)(2) = 64(2) = 128; this is between 120 and 130.<\/p>\n<p>Question 56.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nA landscaper has 125 tiles to build a square patio. The patio must have an area of at least 80 square feet.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16968\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-22.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 22\" width=\"158\" height=\"144\" \/><br \/>\na. What are the possible arrangements for the patio?<\/p>\n<p>Answer:<br \/>\nGiven that a square patio of at least 80 square feet is to be built from 125 tiles of length 12 inches or 1 foot.<br \/>\nSince there are 125 tiles and the patio has a shape of a square of at least 80 square feet, then the possible dimensions of the patio are<br \/>\n9 ft \u00d7 9 ft = 81 ft<br \/>\n10 ft \u00d7 10 ft = 100 ft, and<br \/>\n11 ft \u00d7 11 ft = 121 ft.<\/p>\n<p>b. How many tiles are not used in each arrangement?<\/p>\n<p>Answer:<br \/>\nFor a patio of dimensions, 9ft by 9ft, the number of tiles that will not be used is given by 125 &#8211; 81 = 44<br \/>\nFor a patio of dimensions, 10ft by 10ft, the number of tiles that will not be used is given by 125 &#8211; 100 = 25<br \/>\nFor a patio of dimensions, 11ft by 11ft, the number of tiles that will not be used is given by 125 &#8211; 121 = 4<\/p>\n<p>Question 57.<br \/>\n<strong>PATTERNS<\/strong><br \/>\nCopy and complete the table. Describe what happens to the value of the power as the exponent decreases. Use this pattern to \ufb01nd the value of 4<sup>0<\/sup>.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16969\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 23\" width=\"335\" height=\"80\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23.png 335w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23-300x72.png 300w\" sizes=\"(max-width: 335px) 100vw, 335px\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-22407\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23-300x72.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23\" width=\"300\" height=\"72\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23-300x72.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-23.jpg 335w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\n4<sup>0<\/sup> = 1<br \/>\nThus the value of 4<sup>0<\/sup> is 1.<\/p>\n<p>Question 58.<br \/>\n<strong>REPEATED REASONING<\/strong><br \/>\nHow many blocks do you need to add to Square 6 to get Square 7? to Square 9 to get Square 10? to Square 19 to get Square 20? Explain.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16970\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-24.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 24\" width=\"451\" height=\"152\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-24.png 451w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-24-300x101.png 300w\" sizes=\"(max-width: 451px) 100vw, 451px\" \/><\/p>\n<p>Answer:<br \/>\nYou need to add 14 blocks to get square 7. The square 7 contains 7 \u00d7 7 = 49 blocks<br \/>\nYou need to add 32 blocks to get square 9. The square 9 contains 9 \u00d7 9 = 81 blocks<br \/>\nYou need to add 19 blocks to get square 10. The square 10 contains 10 \u00d7 10 = 100 blocks<br \/>\nYou need to add 261 blocks to get square 19. The square 19 contains 19 \u00d7 19 = 361 blocks<br \/>\nYou need to add 39 blocks to get square 20. The square 20 contains 20 \u00d7 20 = 400 blocks<\/p>\n<h3><a id=\"Lesson_1.2_Order_of_Operations\"><\/a>Lesson 1.2 Order of Operations<\/h3>\n<p><strong>Order of Operations<\/strong><\/p>\n<p><strong>EXPLORATION 1<\/strong><br \/>\n<strong>Comparing Different Orders<\/strong><\/p>\n<p><strong>Work with a partner. Find the value of each expression by using different orders of operations. Are your answers the same?<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-16971\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-25.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 25\" width=\"406\" height=\"241\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-25.png 406w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-25-300x178.png 300w\" sizes=\"(max-width: 406px) 100vw, 406px\" \/><\/p>\n<p>Answer:<br \/>\nThe answers for all the expressions are not the same. The values of each expression will change if you change the order of operations.<br \/>\na. 3 + 2 \u00d7 2<br \/>\n5 \u00d7 2 = 10<br \/>\nMultiply, then add<br \/>\n3 + 2 \u00d7 2<br \/>\n3 + 4 = 7<br \/>\nb. Subtract then multiply<br \/>\n18 &#8211; 3 \u00d7 3<br \/>\n15 \u00d7 3 = 45<br \/>\nMultiply, then subtract<br \/>\n18 &#8211; 3 \u00d7 3<br \/>\n18 &#8211; 9 = 9<br \/>\nc. Multiply, then subtract<br \/>\n8 \u00d7 8 &#8211; 2<br \/>\n64 &#8211; 2 = 62<br \/>\nSubtract, then Multiply<br \/>\n8 \u00d7 8 &#8211; 2<br \/>\n8 \u00d7 6 = 48<br \/>\nd. Multiply, then add<br \/>\n6 \u00d7 6 + 2<br \/>\n36 + 2 = 38<br \/>\nAdd, then multiply<br \/>\n6 \u00d7 6 + 2<br \/>\n6 \u00d7 8 = 48<\/p>\n<p><strong>EXPLORATION 2<\/strong><br \/>\n<strong>Determining Order of Operations<\/strong><br \/>\n<strong>Work with a partner.<\/strong><br \/>\na. Scienti\ufb01c calculators use a standard order of operations when evaluating expressions. Why is a standard order of operations needed?<\/p>\n<p>Answer: The order of operations is a rule that tells you the right order in which to solve different parts of a math problem. The order of operations is important because it guarantees that people can all read and solve a problem in the same way.<\/p>\n<p>b. Use a scienti\ufb01c calculator to evaluate each expression in Exploration 1. Enter each expression exactly as written. For each expression, which order of operations is correct?<\/p>\n<p>Answer:<br \/>\na. 3 + 2 \u00d7 2\u00a0 &#8211; Multiply, then add<br \/>\nb. 18 &#8211; 3 \u00d7 3 &#8211; Multiply, then subtract<br \/>\nc. 8 \u00d7 8 &#8211; 2 &#8211; Multiply, then subtract<br \/>\nd. 6 \u00d7 6 + 2 &#8211; Multiply, then add<\/p>\n<p>c. What order of operations should be used to evaluate 3 + 2<sup>2<\/sup>, 18 \u2212 3<sup>2<\/sup>, 8<sup>2<\/sup> \u2212 2, and 6<sup>2<\/sup> + 2?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16972\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-26.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 26\" width=\"291\" height=\"207\" \/><\/p>\n<p>Answer:<br \/>\nSolve the expressions by using the calculator.<br \/>\na. 3 + 2 \u00d7 2<br \/>\n3 + 4 = 7<br \/>\nb. 18 &#8211; 3 \u00d7 3<br \/>\n18 &#8211; 9 = 9<br \/>\nc. 8 \u00d7 8 &#8211; 2<br \/>\n64 &#8211; 2 = 62<br \/>\nd. 6 \u00d7 6 + 2<br \/>\n36 + 2 = 38<br \/>\nd. Do 18 \u00f7 3.3 and 18 \u00f7 3<sup>2<\/sup> have the same value? Justify your answer.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16973\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-27.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 27\" width=\"153\" height=\"178\" \/><\/p>\n<p>Answer: No<\/p>\n<p>Explanation:<br \/>\n18 \u00f7 3.3<br \/>\n(18 \u00f7 3) \u00d7 3<br \/>\n6 \u00d7 3 = 18<br \/>\n18 \u00f7 3<sup>2<\/sup> \u00a0= 2<br \/>\nBy using the calculator you can find the difference.<br \/>\ne. How does evaluating powers \ufb01t into the order of operations?<\/p>\n<p>Answer:<br \/>\nWhen an expression has parentheses and powers, evaluate it in the following order: contents of parentheses, powers from left to right, multiplication and division from left to right, and addition and subtraction from left to right.<\/p>\n<p><strong>1.2 Lesson<\/strong><\/p>\n<p>A numerical expression is an expression that contains numbers and operations. To evaluate, or \ufb01nd the value of, a numerical expression, use a set of rules called the order of operations.<br \/>\n<strong>Key Idea<\/strong><br \/>\norder of operations<\/p>\n<ol>\n<li>Perform operations in grouping symbols.<\/li>\n<li>Evaluate numbers with exponents.<\/li>\n<li>Multiply and divide from left to right.<\/li>\n<li>Add and subtract from left to right.<\/li>\n<\/ol>\n<p><strong>Try It<\/strong><br \/>\na. Evaluate the expression.<\/p>\n<p>Question 1.<br \/>\n7.5 + 3<\/p>\n<p>Answer: 56<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n7(5 + 3) = 7 \u00d7 8<br \/>\n= 56<\/p>\n<p>Question 2.<br \/>\n(28 &#8211; 20) \u00f7 4<\/p>\n<p>Answer: 2<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n28 &#8211; 20 = 8<br \/>\n8 \u00f7 4 = 2<\/p>\n<p>Question 3.<br \/>\n[6 + (15 &#8211; 10)] \u00d7 5<\/p>\n<p>Answer: 55<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n[6 + (15 &#8211; 10)] \u00d7 5<br \/>\n[6 + 5] \u00d7 5<br \/>\n11 \u00d7 5 = 55<\/p>\n<p><strong>Try It<\/strong><br \/>\nEvaluate the expression.<\/p>\n<p>Question 4.<br \/>\n6 + 2<sup>4<\/sup> &#8211; 1<\/p>\n<p>Answer: 21<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n6 + 2<sup>4<\/sup> &#8211; 1<br \/>\n6 + (16 &#8211; 1)<br \/>\n6 + 15 = 21<br \/>\n6 + 2<sup>4<\/sup> &#8211; 1 = 21<\/p>\n<p>Question 5.<br \/>\n4.3<sup>2<\/sup> + 18 &#8211; 9<\/p>\n<p>Answer: 45<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n4.3<sup>2<\/sup> + (18 &#8211; 9)<br \/>\n4.3<sup>2<\/sup> + 9<br \/>\n4 \u00d7 9 + 9<br \/>\n36 + 9 = 45<\/p>\n<p>Question 6.<br \/>\n16 + (5<sup>2<\/sup> &#8211; 7) \u00f7 3<\/p>\n<p>Answer:<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n16 + (5<sup>2<\/sup> &#8211; 7) \u00f7 3<br \/>\n16 + (25 &#8211; 7) \u00f7 3<br \/>\n16 + (18) \u00f7 3<br \/>\n16 + (18 \u00f7 3)<br \/>\n16 + 6 = 22<\/p>\n<p>Thee symbols \u00d7 and . are used to indicate multiplication. You can also use parentheses to indicate multiplication. For example, 3(2 +7) is the same as 3 \u00d7 (2 + 7).<\/p>\n<p><strong>Try It<\/strong><br \/>\nEvaluate the expression.<\/p>\n<p>Question 7.<br \/>\n50 + 6(12 \u00f7 4) &#8211; 8<sup>2<\/sup><\/p>\n<p>Answer: 4<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n50 + 6(12 \u00f7 4) &#8211; 8<sup>2<\/sup><br \/>\n50 + 6(3) &#8211; 8<sup>2<\/sup><br \/>\n50 + 18 &#8211; 8<sup>2<\/sup><br \/>\n50 + 18 &#8211; 64<br \/>\n68 &#8211; 64<br \/>\n4<\/p>\n<p>Question 8.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16974\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-28.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 28\" width=\"112\" height=\"52\" \/><\/p>\n<p>Answer: 24<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n5\u00b2 &#8211; 1\/5 (10 &#8211; 5)<br \/>\n5\u00b2 &#8211; 1\/5 (5)<br \/>\n5\u00b2 &#8211; 1<br \/>\n25 &#8211; 1<br \/>\n24<\/p>\n<p>Question 9.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16975\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-29.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 29\" width=\"64\" height=\"47\" \/><\/p>\n<p>Answer:<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n8(2+5) = 8 \u00d7 7<br \/>\n(8 \u00d7 7)\/7 = 8<\/p>\n<p><strong>Self-Assessment for Concepts &amp; Skills<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p><strong>USING ORDER OF OPERATIONS<\/strong><br \/>\n<strong>Evaluate the expression.<\/strong><\/p>\n<p>Question 10.<br \/>\n7 + 2.4<\/p>\n<p>Answer: 15<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n7 + 2 \u00d7 4 = 7 + 8 = 15<\/p>\n<p>Question 11.<br \/>\n8 \u00f7 4 \u00d7 2<\/p>\n<p>Answer: 4<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n8 \u00f7 4 = 2<br \/>\n2 \u00d7 2 = 4<\/p>\n<p>Question 12.<br \/>\n3(5 + 1) \u00f7 3<sup>2<\/sup><\/p>\n<p>Answer: 2<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate the expression from left to right.<br \/>\n3(5 + 1) \u00f7 3<sup>2<\/sup><br \/>\n3 \u00d7 6 \u00f7\u00a03<sup>2<\/sup><br \/>\n18 \u00f7 9 =2<\/p>\n<p>Question 13.<br \/>\n<strong>WRITING<\/strong><br \/>\nWhy does 12 \u2212 8 \u00f7 2 = 2?<\/p>\n<p>Answer:<br \/>\n12 \u2212 8 \u00f7 2<br \/>\n4 \u00f7 2 = 2<\/p>\n<p>Question 14.<br \/>\n<strong>REASONING<\/strong><br \/>\nDescribe the steps in evaluating the expression 8 \u00f7 (6 \u2212 4) + 3<sup>2<\/sup>.<\/p>\n<p>Answer:<br \/>\n8 \u00f7 (6 \u2212 4) + 3<sup>2<\/sup><br \/>\n8 \u00f7 2 + 3<sup>2<\/sup><br \/>\n4 + 9 = 13<\/p>\n<p>Question 15.<br \/>\n<strong>WHICH ONE DOESN\u2019T BELONG?<\/strong><br \/>\nWhich expression does not belong with the other three? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16976\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-30.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 30\" width=\"484\" height=\"50\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-30.png 484w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-30-300x31.png 300w\" sizes=\"(max-width: 484px) 100vw, 484px\" \/><\/p>\n<p>Answer: (5<sup>2<\/sup> &#8211; 8) \u00d7 2 does not belong to the other three. Because the order of operations and expressions are different for the fourth option.<\/p>\n<p><strong>Self-Assessment for Problem Solving<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p>Question 16.<br \/>\nA square plot of land has side lengths of 40 meters. An archaeologist divides the land into 64 equal parts. What is the area of each part?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nA square plot of land has side lengths of 40 meters.<br \/>\nAn archaeologist divides the land into 64 equal parts.<br \/>\nSide of the square field = 40m<br \/>\nArea of the square field = s \u00d7 s<br \/>\nA = 40m \u00d7 40m<br \/>\nA = 1600 sq.m<br \/>\nArea of each part of the square = 1600\/64 = 25 sq.m<\/p>\n<p>Question 17.<br \/>\nA glass block window is made of two different-sized glass squares. The window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches. Find the total area of the small glass squares.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16977\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-31.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 31\" width=\"209\" height=\"196\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA glass block window is made of two different-sized glass squares.<br \/>\nThe window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches.<br \/>\n40 \u00d7 10 = 400<\/p>\n<p>Question 18.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nA square vegetable garden has side lengths of 12 feet. You plant \ufb02owers in the center portion as shown. You divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers. What is the area of the onion section?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16978\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-32.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 32\" width=\"174\" height=\"155\" \/><\/p>\n<p>Answer:<br \/>\nA square vegetable garden has side lengths of 12 feet.<br \/>\nYou plant flowers in the center portion of the garden, a square that has side lengths of 4 feet.<br \/>\nYou divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers.<br \/>\nGiven that,<br \/>\n\u2192 side of the square vegetable garden is = 12 feet.<br \/>\nSo,<br \/>\n\u2192 Area of square vegetable garden = (side)\u00b2 = (12)\u00b2 = 144 feet\u00b2.<br \/>\nNow, given that, inside this area, there is a square of side 4 feet reserved for flowers.<br \/>\nSo,<br \/>\n\u2192 The area of the flower section = (side)\u00b2 = (4)\u00b2 = 16 feet\u00b2.<br \/>\nTherefore,<br \/>\n\u2192 The rest of the garden that is intended for vegetables is = The total garden area &#8211; The flower section area = 144 &#8211; 16 = 128 feet\u00b2.<br \/>\nNow, this remaining area is to be divided into four equal sections.<br \/>\nSo,<br \/>\n\u2192 The area of the onion section = (1\/4) of remaining area = (1\/4) \u00d7 128 = 32 feet\u00b2.<\/p>\n<h3><a id=\"Order_of_Operations_Practice_1.2\"><\/a>Order of Operations Practice 1.2<\/h3>\n<p><strong>Review &amp; Refresh<\/strong><\/p>\n<p>Write the product as a power.<\/p>\n<p>Question 1.<br \/>\n11 \u00d7 11 \u00d7 11 \u00d7 11<\/p>\n<p>Answer: The exponent for the product 11 \u00d7 11 \u00d7 11 \u00d7 11 is 11<sup>4<\/sup><\/p>\n<p>Question 2.<br \/>\n13 \u00d7 13 \u00d7 13 \u00d7 13 \u00d7 13<\/p>\n<p>Answer: The exponent for the product 13 \u00d7 13 \u00d7 13 \u00d7 13 \u00d7 13 is 13<sup>5<\/sup><br \/>\nFind the missing dimension of the rectangular prism.<\/p>\n<p>Question 3.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16979\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-33.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 33\" width=\"141\" height=\"155\" \/><\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nl = 6 in.<br \/>\nb = 4 in.<br \/>\nh = ?<br \/>\nv = 192 cu. in<br \/>\nVolume of the rectangular prism = lbh<br \/>\n192 = 6 \u00d7 4 \u00d7 h<br \/>\nh = 192\/24<br \/>\nh = 8<br \/>\nThus the height of the rectangular prism is 8 inches.<\/p>\n<p>Question 4.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16980\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-34.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 34\" width=\"130\" height=\"164\" \/><\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nh = 9m<br \/>\nb = 3m<br \/>\nv = 135 cu. m<br \/>\nl = ?<br \/>\nThe volume of the rectangular prism = lbh<br \/>\n135 = l \u00d7 3 \u00d7 9<br \/>\n135 = l \u00d7 27<br \/>\nl = 5m<\/p>\n<p>Tell whether the number is prime or composite.<\/p>\n<p>Question 5.<br \/>\n9<\/p>\n<p>Answer: Composite Number<br \/>\nA natural number greater than 1 that is not prime is called a composite number.<\/p>\n<p>Question 6.<br \/>\n11<\/p>\n<p>Answer: Prime Number<br \/>\nA prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.<\/p>\n<p>Question 7.<br \/>\n23<\/p>\n<p>Answer: Prime Number<br \/>\nA prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.<\/p>\n<p><strong>Concepts, Skills, &amp; Problem Solving<\/strong><\/p>\n<p><strong>COMPARING DIFFERENT ORDERS<\/strong><br \/>\n<strong>Find the value of the expression by using different orders of operations. Are your answers the same?<\/strong> (See Exploration 1, p. 9.)<\/p>\n<p>Question 8.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16981\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-35.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 35\" width=\"237\" height=\"52\" \/><\/p>\n<p>Answer:<br \/>\n4 + 6 \u00d7 6<br \/>\n10 \u00d7 6 = 60<br \/>\n4 + 6 \u00d7 6<br \/>\n4 + 36 = 40<\/p>\n<p>Question 9.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16982\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-36.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 36\" width=\"285\" height=\"56\" \/><\/p>\n<p>Answer:<br \/>\n5 \u00d7 5 &#8211; 3<br \/>\n5 \u00d7 2 = 10<br \/>\n5 \u00d7 5 &#8211; 3<br \/>\n25 &#8211; 3 = 22<\/p>\n<p><strong>USING ORDER OF OPERATIONS<\/strong><br \/>\n<strong>Evaluate the expression.<\/strong><\/p>\n<p>Question 10.<br \/>\n5 + 18 \u00f7 6<\/p>\n<p>Answer: 8<\/p>\n<p>Explanation:<br \/>\nFirst, divide then divide.<br \/>\n5 + 3 = 8<\/p>\n<p>Question 11.<br \/>\n(11 &#8211; 3) \u00f7 2 + 1<\/p>\n<p>Answer:<\/p>\n<p>Explanation:<br \/>\nFirst, subtract and divide.<br \/>\n(11 &#8211; 3) \u00f7 2 + 1<br \/>\n8 \u00f7 2 + 1<br \/>\n4 + 1 = 5<\/p>\n<p>Question 12.<br \/>\n45 \u00f7 9 \u00d7 2<\/p>\n<p>Answer: 10<\/p>\n<p>Explanation:<br \/>\nThe first divide then multiply.<br \/>\n45 \u00f7 9 \u00d7 2<br \/>\n5 \u00d7 2 = 10<\/p>\n<p>Question 13.<br \/>\n6<sup>2<\/sup> &#8211; 3.4<\/p>\n<p>Answer: 24<\/p>\n<p>Explanation:<br \/>\nMultiply then subtract<br \/>\n6<sup>2<\/sup> &#8211; 3.4<br \/>\n36 &#8211; 12<br \/>\n24<\/p>\n<p>Question 14.<br \/>\n42 \u00f7 (15 &#8211; 2<sup>3<\/sup>)<\/p>\n<p>Answer: 6<\/p>\n<p>Explanation:<br \/>\nSubtract then divide.<br \/>\n42 \u00f7 (15 &#8211; 8)<br \/>\n42 \u00f7 7<br \/>\n6<\/p>\n<p>Question 15.<br \/>\n4<sup>2<\/sup>.2 + 8.7<\/p>\n<p>Answer: 88<\/p>\n<p>Explanation:<br \/>\nMultiply then add<br \/>\n4<sup>2<\/sup>.2 + 8.7<br \/>\n16 \u00d7 2 + 56<br \/>\n32 + 56 = 88<\/p>\n<p>Question 16.<br \/>\n(5<sup>2<\/sup> &#8211; 2) \u00d7 1<sup>5<\/sup> + 4<\/p>\n<p>Answer: 27<\/p>\n<p>Explanation:<br \/>\n(5<sup>2<\/sup> &#8211; 2) \u00d7 1<sup>5<\/sup> + 4<br \/>\n(25 &#8211; 2) \u00d7 1 + 4<br \/>\nAdd, subtract then multiply<br \/>\n23 + 4 = 27<\/p>\n<p>Question 17.<br \/>\n4 + 2 \u00d7 3<sup>2<\/sup> &#8211; 9<\/p>\n<p>Answer: 13<\/p>\n<p>Explanation:<br \/>\n4 + 2 \u00d7 3<sup>2<\/sup> &#8211; 9<br \/>\n4 + 18 &#8211; 9<br \/>\n4 + 9 = 13<\/p>\n<p>Question 18.<br \/>\n8 \u00f7 2 \u00d7 3 + 4<sup>2<\/sup> \u00f7 4<\/p>\n<p>Answer: 16<\/p>\n<p>Explanation:<br \/>\n8 \u00f7 2 \u00d7 3 + 4<sup>2<\/sup> \u00f7 4<br \/>\n(4 \u00d7 3) + (16 \u00f7 4)<br \/>\n12 + 4<br \/>\n16<\/p>\n<p>Question 19.<br \/>\n3<sup>2<\/sup> + 12 \u00f7 (6 &#8211; 3) \u00d7 8<\/p>\n<p>Answer: 41<\/p>\n<p>Explanation:<br \/>\n3<sup>2<\/sup> + 12 \u00f7 (6 &#8211; 3) \u00d7 8<br \/>\n9 + (12 \u00f7 (6 &#8211; 3)) \u00d7 8<br \/>\n9 + (12 \u00f7 3) \u00d7 8<br \/>\n9 + 4 \u00d7 8<br \/>\n9 + 32<br \/>\n41<\/p>\n<p>Question 20.<br \/>\n(10 + 4) \u00f7 (26 &#8211; 19)<\/p>\n<p>Answer: 2<\/p>\n<p>Explanation:<br \/>\nAdd, subtract then divide<br \/>\n(10 + 4) \u00f7 (26 &#8211; 19)<br \/>\n14 \u00f7 7<br \/>\n2<\/p>\n<p>Question 21.<br \/>\n(5<sup>2<\/sup> &#8211; 4).2 &#8211; 18<\/p>\n<p>Answer: 24<\/p>\n<p>Explanation:<br \/>\n((5<sup>2<\/sup> &#8211; 4).2) &#8211; 18<br \/>\n((25 &#8211; 4) \u00d7 2) &#8211; 18<br \/>\n(21 \u00d7 2) &#8211; 18<br \/>\n42 &#8211; 18<br \/>\n24<\/p>\n<p>Question 22.<br \/>\n2 \u00d7 [(16 &#8211; 8) \u00d7 2]<\/p>\n<p>Answer: 32<\/p>\n<p>Explanation:<br \/>\n2 \u00d7 [(16 &#8211; 8) \u00d7 2]<br \/>\n2 \u00d7 [8 \u00d7 2]<br \/>\n2 \u00d7 16<br \/>\n32<\/p>\n<p>Question 23.<br \/>\n12 + 8 \u00d7 3<sup>3<\/sup> &#8211; 24<\/p>\n<p>Answer: 204<\/p>\n<p>Explanation:<br \/>\n12 + 8 \u00d7 3<sup>3<\/sup> &#8211; 24<br \/>\n12 + (8 \u00d7 27) &#8211; 24<br \/>\n12 + 216 &#8211; 24<br \/>\n12 + 192 = 204<\/p>\n<p>Question 24.<br \/>\n6<sup>2<\/sup> \u00f7 [(2 + 4) \u00d7 2<sup>3<\/sup>]<\/p>\n<p>Answer: 48<\/p>\n<p>Explanation:<br \/>\n6<sup>2<\/sup> \u00f7 [(2 + 4) \u00d7 2<sup>3<\/sup>]<br \/>\n36 \u00f7 [(2 + 4) \u00d7 2<sup>3<\/sup>]<br \/>\n36 \u00f7 6 \u00d7 8<br \/>\n6 \u00d7 8<br \/>\n48<\/p>\n<p><strong>YOU BE THE TEACHER<br \/>\n<\/strong><strong>Your friend evaluates the expression. Is your friend correct? Explain your reasoning.<\/strong><\/p>\n<p>Question 25.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17008\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-37.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 37\" width=\"229\" height=\"99\" \/><\/p>\n<p>Answer: Your friend is incorrect.<br \/>\n9 + 3 \u00d7 3\u00b2<br \/>\n9 + (27)<br \/>\n36<\/p>\n<p>Question 26.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17009\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-38.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 38\" width=\"234\" height=\"109\" \/><\/p>\n<p>Answer:<br \/>\n19 &#8211; 6 + 12<br \/>\n13 + 12<br \/>\n25<\/p>\n<p>Question 27.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nYou need to read 20 poems in 5 days for an English project. Each poem is 2 pages long. Evaluate the expression 20 \u00d7 2 \u00f7 5 to \ufb01nd how many pages you need to read each day.<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nYou need to read 20 poems in 5 days for an English project. Each poem is 2 pages long.<br \/>\n20 \u00d7 2 \u00f7 5<br \/>\n40 \u00f7 5 = 8<br \/>\nThus you need to read 8 pages each day.<\/p>\n<p><strong>USING ORDER OF OPERATIONS<\/strong><br \/>\n<strong>Evaluate the expression.<\/strong><\/p>\n<p>Question 28.<br \/>\n12 &#8211; 2(7 &#8211; 4)<\/p>\n<p>Answer:<br \/>\n12 -(2 \u00d7 (7 &#8211; 4))<br \/>\n12 &#8211; (2 \u00d7 3)<br \/>\n12 &#8211; 6 = 6<\/p>\n<p>Question 29.<br \/>\n4(3 + 5) &#8211; 3(6 -2)<\/p>\n<p>Answer:<br \/>\n4(3 + 5) &#8211; 3(6 -2)<br \/>\n4 \u00d7 8 &#8211; 3 \u00d7 4<br \/>\n32 &#8211; 12<br \/>\n20<\/p>\n<p>Question 30.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17010\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-39.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 39\" width=\"105\" height=\"46\" \/><\/p>\n<p>Answer:<br \/>\n6 + 1\/4 (12 -8)<br \/>\n6 + 1\/4(4)<br \/>\n6 + 1<br \/>\n7<\/p>\n<p>Question 31.<br \/>\n9<sup>2<\/sup> &#8211; 8(6 + 2)<\/p>\n<p>Answer:<br \/>\n81 &#8211; (8(6 + 2))<br \/>\n81 &#8211; (8 \u00d7 8)<br \/>\n81 &#8211; 64<br \/>\n17<\/p>\n<p>Question 32.<br \/>\n4(3 &#8211; 1)<sup>3<\/sup> + 7(6) &#8211; 5<sup>2<\/sup><\/p>\n<p>Answer:<br \/>\n4(3 &#8211; 1)<sup>3<\/sup> + 7(6) &#8211; 5<sup>2<\/sup><br \/>\n4(2)<sup>3<\/sup> + 7(6) &#8211; 5<sup>2<\/sup><br \/>\n4 \u00d7 8 + 42 &#8211; 25<br \/>\n32 + 42 &#8211; 25 = 49<\/p>\n<p>Question 33.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17011\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-40.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 40\" width=\"127\" height=\"55\" \/><\/p>\n<p>Answer:<br \/>\n8[(1 1\/6 + 5\/6) \u00f7 4]<br \/>\n[8[7\/6 + 5\/6] \u00f7 4]<br \/>\n8[12\/6] \u00f7 4<br \/>\n8[2 \u00f7 4]<br \/>\n8(1\/2)<br \/>\n4<\/p>\n<p>Question 34.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17012\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-41.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 41\" width=\"113\" height=\"54\" \/><\/p>\n<p>Answer:<br \/>\n49 &#8211; 2((11-3)\/8)<br \/>\n49 &#8211; 2 (8\/8)<br \/>\n49 &#8211; 2<br \/>\n47<\/p>\n<p>Question 35.<br \/>\n8(7.3 + 3.7 &#8211; 8) \u00f7 2<\/p>\n<p>Answer:<br \/>\n8(7.3 + 3.7 &#8211; 8) \u00f7 2<br \/>\n(8(7.3 + 3.7 &#8211; 8)) \u00f7 2<br \/>\n8 (11 &#8211; 8) \u00f7 2<br \/>\n8 \u00d7 3 \u00f7 2<br \/>\n24 \u00f7 2<br \/>\n12<\/p>\n<p>Question 36.<br \/>\n2<sup>4<\/sup>(5.2 &#8211; 3.2) \u00f7 4<\/p>\n<p>Answer:<br \/>\n2<sup>4<\/sup>(5.2 &#8211; 3.2) \u00f7 4<br \/>\n16 (5.2 &#8211; 3.2) \u00f7 4<br \/>\n16 (2) \u00f7 4<br \/>\n32 \u00f7 4<br \/>\n8<\/p>\n<p>Question 37.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17013\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-42.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 42\" width=\"76\" height=\"49\" \/><\/p>\n<p>Answer:<br \/>\n36(3+5)\/4<br \/>\n36 \u00d7 8\/4<br \/>\n36 \u00d7 2<br \/>\n72<\/p>\n<p>Question 38.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17014\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-43.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 43\" width=\"106\" height=\"56\" \/><\/p>\n<p>Answer:<br \/>\n(144 &#8211; 24 + 1)\/121<br \/>\n121\/121<br \/>\n1<\/p>\n<p>Question 39.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17015\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-44.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 44\" width=\"89\" height=\"53\" \/><\/p>\n<p>Answer:<br \/>\n26 \u00f7 2 + 5 = 18<br \/>\n18\/6 = 3<\/p>\n<p>Question 40.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nBefore a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave. Evaluate the expression 8 + 5(4) \u2212 3(2) to \ufb01nd how many people are in the theater.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17016\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-45.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 45\" width=\"128\" height=\"252\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nBefore a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave.<br \/>\n8 + (5 \u00d7 4) &#8211; (3 \u00d7 2)<br \/>\n8 + 20 &#8211; 6<br \/>\n28 &#8211; 6<br \/>\n22<\/p>\n<p>Question 41.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nThe front door of a house is painted white and blue. Each window is a square with a side length of 7 inches. What is the area of the door that is painted blue?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nThe front door of a house is painted white and blue. Each window is a square with a side length of 7 inches.<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 7 in \u00d7 7 in<br \/>\nA = 49 sq. inches<br \/>\nTherefore the area of the door that is painted blue is 49 sq. inches.<\/p>\n<p>Question 42.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nYou buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school. After using a $10 gift card, how much do you owe? Explain how you solved the problem.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17017\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-46.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 46\" width=\"255\" height=\"204\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nYou buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school.<br \/>\nCost of 1 notebook = $2<br \/>\n6 notebooks = 6 \u00d7 $2 = $12<br \/>\nCost of 1 folder = $1<br \/>\n10 folders = 10 \u00d7 $1 = $10<br \/>\nCost of 1 pack of pencils = $3<br \/>\nCost of 1 lunch box = $8<br \/>\nSo the total cost is $11 + $10 + $3 + $8 = $31<br \/>\nYou used $10 gift card.<br \/>\n$31 &#8211; $10 = $21<br \/>\nThus you ow $21.<\/p>\n<p>Question 43.<br \/>\n<strong>OPEN-ENDED<\/strong><br \/>\nUse all four operations and at least one exponent to write an expression that has a value of 100.<\/p>\n<p>Answer: You need to use +, -, \u00d7, \u00f7\u00a0 operations to write the expressions that have the value of 100.<br \/>\n(34 &#8211; 1) \u00d7 3 + 3\u00b2 \u00f7 9 = 100<\/p>\n<p>Question 44.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-17018\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-47.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 47\" width=\"258\" height=\"203\" \/><\/p>\n<p><strong>REPEATED REASONING<\/strong><br \/>\nA Petri dish contains 35 cells. Every day, each cell in the Petri dish divides into 2 cells in a process called mitosis. How many cells are there after 14 days? Justify your answer.<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA Petri dish contains 35 cells. Every day, each cell in the Petri dish divides into 2 cells in a process called mitosis<br \/>\n35 \u00f7 2 = 17.5<br \/>\n1 day = 0.5 + 0.5 cells<br \/>\n14 days = 14 \u00d7 1 = 14 cells<br \/>\n17.5 &#8211; 14 = 3.5<br \/>\nThus there are 3.5 cells after 14 days.<\/p>\n<p>Question 45.<br \/>\n<strong>REASONING<\/strong><br \/>\nTwo groups collect litter along the side of a road. It takes each group 5 minutes to clean up a 200-yard section. How long does it take both groups working together to clean up 2 miles? Explain how you solved the problem.<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nTwo groups collect litter along the side of a road. It takes each group 5 minutes to clean up a 200-yard section.<br \/>\nTo convert 2 miles to yards, you have to multiply 2 by 1760, because 1 mile equals to 1760 yards:<br \/>\n2 \u00d7 1760 = 3520 yards.<br \/>\nIf you would like to know how long does it take to clean up 2 miles, you can calculate this using the following steps:<br \/>\n5 \u00d7 3520 = 200 \u00d7 x<br \/>\n17600 = 200 \u00d7 x \/200<br \/>\nx = 17600 \/ 200<br \/>\nx = 88 minutes<\/p>\n<p>Question 46.<br \/>\n<strong>NUMBER SENSE<\/strong><br \/>\nCopy each statement. Insert +, \u2212, \u00d7, or \u00f7 symbols to make each statement true.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17019\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 48\" width=\"571\" height=\"81\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48.png 571w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48-300x43.png 300w\" sizes=\"(max-width: 571px) 100vw, 571px\" \/><\/p>\n<p>Answer:<br \/>\nYou can find the value by using the calculator by inserting the suitable operations.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-23518\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48-300x43.jpg\" alt=\"Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48\" width=\"300\" height=\"43\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48-300x43.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-48.jpg 571w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h3><a id=\"Lesson_1.3_Prime_Factorization\"><\/a>Lesson 1.3 Prime Factorization<\/h3>\n<p>Prime Factorization<\/p>\n<p><strong>EXPLORATION 1<\/strong><br \/>\n<strong>Rewriting Numbers as Products of Factors<\/strong><\/p>\n<p><strong>Work with a partner. Two students use factor trees to write 108 as a product of factors, as shown below.<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-17020\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-49.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 49\" width=\"519\" height=\"330\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-49.png 519w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-49-300x191.png 300w\" sizes=\"(max-width: 519px) 100vw, 519px\" \/><br \/>\na. Without using 1 as a factor, can you write 108 as a product with more factors than each student used? Justify your answer. Math Practice<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17022\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-50.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 50\" width=\"330\" height=\"328\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-50.png 330w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-50-300x298.png 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-50-150x150.png 150w\" sizes=\"(max-width: 330px) 100vw, 330px\" \/><\/p>\n<p>Answer: Yes you can find the factors by using the prime factorization.<br \/>\n108 = 2 \u00d7 54<br \/>\n= 2 \u00d7 2 \u00d7 27<br \/>\n= 2 \u00d7 2 \u00d7 3 \u00d7 9<br \/>\n= 2 \u00d7 2 \u00d7 3 \u00d7 3 \u00d7 3<\/p>\n<p>b. Use factor trees to write 80, 162, and 300 as products of as many factors as possible. Do not use 1 as a factor.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17021\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-50.1.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 50.1\" width=\"152\" height=\"131\" \/><\/p>\n<p>Answer:<br \/>\n80 = 2 \u00d7 40<br \/>\n= 2 \u00d7 2 \u00d7 20<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 10<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 5<br \/>\n162 = 2 \u00d7 81<br \/>\n= 2 \u00d7 3 \u00d7 27<br \/>\n= 2 \u00d7 3 \u00d7 3 \u00d7 9<br \/>\n= 2 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3<br \/>\n300 = 2 \u00d7 150<br \/>\n= 2 \u00d7 2 \u00d7 75<br \/>\n= 2 \u00d7 2 \u00d7 3 \u00d7 25<br \/>\n= 2 \u00d7 2 \u00d7 3 \u00d7 5 \u00d7 5<\/p>\n<p>c. Compare your results in parts (a) and (b) with other groups. For each number, identify the product with the greatest number of factors. What do these factors have in common?<\/p>\n<p>Answer: 300 contains the greatest number of factors. (2, 2, 3, 5, 5)<\/p>\n<p><strong>1.3 Lesson<\/strong><\/p>\n<p>Because 2 is a factor of 10 and 2 . 5 =10, 5 is also a factor of 10. The pair 2, 5 is called a factor pair of 10.<\/p>\n<p><strong>Try It<\/strong><br \/>\nList the factor pairs of the number.<\/p>\n<p>Question 1.<br \/>\n18<\/p>\n<p>Answer: The factor pairs of 18 are 1, 2, 3, 6, 9, 18<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 18 = 18<br \/>\n2 \u00d7 9 = 18<br \/>\n3 \u00d7 6 = 18<br \/>\n6 \u00d7 3 = 18<br \/>\n9 \u00d7 2 = 18<br \/>\n18 \u00d7 1 = 18<\/p>\n<p>Question 2.<br \/>\n24<\/p>\n<p>Answer: The factor pairs of 1, 2, 3, 4, 6, 8, 12, 24<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 24 = 24<br \/>\n2 \u00d7 12 = 24<br \/>\n3 \u00d7 8 = 24<br \/>\n4 \u00d7 6 = 24<br \/>\n6 \u00d7 4 = 24<br \/>\n8 \u00d7 3 = 24<br \/>\n12 \u00d7 2 = 24<br \/>\n24 \u00d7 1 = 24<\/p>\n<p>Question 3.<br \/>\n51<\/p>\n<p>Answer: The factor pairs of 1, 3, 17, 51<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 51 = 51<br \/>\n3 \u00d7 17 = 51<br \/>\n17 \u00d7 3 = 51<br \/>\n51 \u00d7 1 = 15<\/p>\n<p>Question 4.<br \/>\n<strong>WHAT IF?<\/strong><br \/>\nThe woodwinds section of the marching band has 38 members. Which has more possible arrangements, the brass section or the woodwinds section? Explain.<\/p>\n<p>Answer: Brass section. 38 has only two-factor pairs.<br \/>\n38 = 1 \u00d7 38<br \/>\n= 2 \u00d7 19<\/p>\n<p><strong>Key Idea<\/strong><\/p>\n<p><strong>Prime Factorization<\/strong><br \/>\nThe prime factorization of a composite number is the number written as a product of its prime factors.<br \/>\nYou can use factor pairs and a factor tree to help \ufb01nd the prime factorization of a number. The factor tree is complete when only prime factors appear in the product. A factor tree for 60 is shown.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17023\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-51.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 51\" width=\"217\" height=\"133\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17024\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-52.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 52\" width=\"190\" height=\"246\" \/><\/p>\n<p><strong>Try It<\/strong><br \/>\nWrite the prime factorization of the number.<\/p>\n<p>Question 5.<br \/>\n20<\/p>\n<p>Answer:<br \/>\nThe Prime Factorization is:<br \/>\n2 x 2 x 5<br \/>\nIn Exponential Form:<br \/>\n2\u00b2 x 51<br \/>\nCSV Format:<br \/>\n2, 2, 5<\/p>\n<p>Question 6.<br \/>\n88<\/p>\n<p>Answer:<br \/>\n88 = 2 \u00d7 44<br \/>\n= 2 \u00d7 2 \u00d7 22<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 11<br \/>\nThe Prime Factorization is: 2 \u00d7 2 \u00d7 2 \u00d7 11<\/p>\n<p>Question 7.<br \/>\n90<\/p>\n<p>Answer:<br \/>\n90 = 2 \u00d7 45<br \/>\n= 2 \u00d7 3 \u00d7 15<br \/>\n= 2 \u00d7 3 \u00d7 3 \u00d7 5<br \/>\nThe Prime Factorization is: 2 \u00d7 3 \u00d7 3 \u00d7 5<\/p>\n<p>Question 8.<br \/>\n462<\/p>\n<p>Answer:<br \/>\n= 2 \u00d7 231<br \/>\n= 2 \u00d7 3 \u00d7 77<br \/>\n= 2 \u00d7 3 \u00d7 7 \u00d7 11<br \/>\nThe Prime Factorization is: 2 \u00d7 3 \u00d7 7 \u00d7 11<\/p>\n<p><strong>Self-Assessment for Concepts &amp; Skills<\/strong><\/p>\n<p><strong>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/strong><\/p>\n<p><strong>WRITING A PRIME FACTORIZATION<\/strong><br \/>\n<strong>Write the prime factorization of the number.<\/strong><\/p>\n<p>Question 9.<br \/>\n14<\/p>\n<p>Answer:<br \/>\n14 = 2 \u00d7 7<br \/>\nThe Prime Factorization is: 2 \u00d7 7<\/p>\n<p>Question 10.<br \/>\n86<\/p>\n<p>Answer:<br \/>\n86 = 2 \u00d7 43<br \/>\nThe Prime Factorization is: 2 \u00d7 43<\/p>\n<p>Question 11.<br \/>\n40<\/p>\n<p>Answer:<br \/>\n40 = 2 \u00d7 20<br \/>\n= 2 \u00d7 2 \u00d7 10<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 5<br \/>\nThe Prime Factorization is: 2 \u00d7 2 \u00d7 2 \u00d7 5<\/p>\n<p>Question 12.<br \/>\n516<\/p>\n<p>Answer:<br \/>\n516 = 2 \u00d7 258<br \/>\n= 2 \u00d7 2 \u00d7 129<br \/>\n= 2 \u00d7 2 \u00d7 3 \u00d7 43<br \/>\nThe Prime Factorization is: 2 \u00d7 2 \u00d7 3 \u00d7 43<\/p>\n<p>Question 13.<br \/>\n<strong>WRITING<\/strong><br \/>\nExplain the difference between prime numbers and composite numbers.<\/p>\n<p>Answer:<br \/>\nA prime number is a number that has exactly two factors i.e. &#8216;1&#8217; and the number itself. A composite number has more than two factors, which means apart from getting divided by number 1 and itself, it can also be divided by at least one integer or number.<\/p>\n<p>Question 14.<br \/>\n<strong>STRUCTURE<\/strong><br \/>\nYour friend lists the following factor pairs and concludes that there are 6 factor pairs of 12. Explain why your friend is incorrect.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17025\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-53.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 53\" width=\"218\" height=\"106\" \/><\/p>\n<p>Answer: Your friend is incorrect. Because there are 5-factor pairs of 12.<br \/>\nThe factor pairs of 12 are<br \/>\n1 \u00d7 12 =12<br \/>\n2 \u00d7 6 = 12<br \/>\n3 \u00d7 4 = 12<br \/>\n4 \u00d7 3 = 12<br \/>\n6 \u00d7 2 = 12<\/p>\n<p>Question 15.<br \/>\n<strong>WHICH ONE DOESN\u2019T BELONG?<\/strong><br \/>\nWhich factor pair does not belong with the other three? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17026\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-54.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 54\" width=\"302\" height=\"54\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-54.png 302w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-54-300x54.png 300w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><\/p>\n<p>Answer:<br \/>\n2, 28 = 2 \u00d7 28 = 56<br \/>\n4, 14 = 4 \u00d7 14 = 56<br \/>\n6, 9 = 6 \u00d7 9 = 54<br \/>\n7, 8 = 7 \u00d7 56<br \/>\nBy this we can say that 6, 9 does not belong to the other three expressions.<\/p>\n<p><strong>Self-Assessment for Problem Solving<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p>Question 16.<br \/>\nA group of 20 friends plays a card game. The game can be played with 2 or more teams of equal size. Each team must have atleast 2 members. List the possible numbers and sizes of teams.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17028\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-55.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 55\" width=\"266\" height=\"170\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA group of 20 friends plays a card game. The game can be played with 2 or more teams of equal size. Each team must have at least 2 members.<br \/>\n20 = 1 \u00d7 20<br \/>\n2 \u00d7 10<br \/>\n4 \u00d7 5<br \/>\n5 \u00d7 4<br \/>\n10 \u00d7 2<br \/>\nThus there are 5 possible numbers and size of teams.<\/p>\n<p>Question 17.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17029\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-56.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 56\" width=\"136\" height=\"266\" \/><br \/>\nYou arrange 150 chairs in rows for a school play. You want each row to have the same number of chairs. How many possible arrangements are there? Are all of the possible arrangements appropriate for the play? Explain.<\/p>\n<p>Answer:<br \/>\nYou arrange 150 chairs in rows for a school play. You want each row to have the same number of chairs.<br \/>\n150 = 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150<br \/>\nThus there are 12 possible arrangements appropriate for the play.<\/p>\n<p>Question 18.<br \/>\nWhat is the least perfect square that is a factor of 4536? What is the greatest perfect square that is a factor of 4536?<\/p>\n<p>Answer: least perfect square that is a factor of 4536<\/p>\n<p>Explanation:<br \/>\nWhat is the last number of 4,536? It is this number: 4536. The answer is 6. Is 6 in the list of numbers that are never perfect squares (2, 3, 7 or 8)?<br \/>\nAnswer: No, 6 is not in the list of numbers that are never perfect squares. Let&#8217;s continue to the next step.<br \/>\nStep 2:<br \/>\nWe now need to obtain the digital root of the number. Here&#8217;s how you do it:<br \/>\nSplit the number up and add each digit together:<br \/>\n4 + 5 + 3 + 6 = 18<br \/>\nIf the answer is more than one digit, you would add each digit of the answer together again:<br \/>\n1 + 8 = 9<br \/>\n1 x 4,5362 x 2,2683 x 1,5124 x 1,1346 x 7567 x 6488 x 5679 x 50412 x 37814 x 32418 x 25221 x 21624 x 18927 x 16828 x 16236 x 12642 x 10854 x 8456 x 8163 x 72<br \/>\nWe&#8217;re looking for a factor combination with equal numbers for X and Y (like 3&#215;3) above. Notice there isn&#8217;t an equal factor combination, that when multiplied together, produce the number 4,536. That means 4,536 is NOT a perfect square.<\/p>\n<p>Question 19.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nThe prime factorization of a number is 2<sup>4<\/sup> \u00d7 3<sup>4<\/sup> \u00d7 5<sup>4<\/sup> \u00d7 7<sup>2<\/sup>. Is the number a perfect square? Explain your reasoning.<\/p>\n<p>Answer:<br \/>\nThe prime factorization of a number is 2<sup>4<\/sup> \u00d7 3<sup>4<\/sup> \u00d7 5<sup>4<\/sup> \u00d7 7<sup>2<\/sup>.<br \/>\n16 \u00d7 81 \u00d7 625 \u00d7 49 = 39690000<br \/>\nYes, 39690000 is a perfect square.<\/p>\n<h3><a id=\"Prime_Factorization_Practice_1.3\"><\/a>Prime Factorization Practice 1.3<\/h3>\n<p><strong>Review &amp; Refresh<\/strong><\/p>\n<p>Evaluate the expression.<\/p>\n<p>Question 1.<br \/>\n2 + 4<sup>2<\/sup>(5 &#8211; 3)<\/p>\n<p>Answer: 34<\/p>\n<p>Explanation:<br \/>\n2 + 4<sup>2<\/sup>(5 &#8211; 3)<br \/>\n2 + 4<sup>2<\/sup>(2)<br \/>\n2 + 16(2) = 2 + 32 = 34<\/p>\n<p>Question 2.<br \/>\n2<sup>3<\/sup> + 4 \u00d7 3<sup>2<\/sup><\/p>\n<p>Answer: 44<\/p>\n<p>Explanation:<br \/>\n2<sup>3<\/sup> + 4 \u00d7 3<sup>2<\/sup><br \/>\n8 + 4 \u00d7 9<br \/>\n8 + 36<br \/>\n44<\/p>\n<p>Question 3.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17030\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-57.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 57\" width=\"135\" height=\"57\" \/><\/p>\n<p>Answer:<\/p>\n<p>Explanation:<br \/>\n9 \u00d7 5 &#8211; 16(5\/2 &#8211; 1\/2)<br \/>\n9 \u00d7 5 &#8211; 16(4\/2)<br \/>\n45 &#8211; 16(2)<br \/>\n45 &#8211; 32<br \/>\n13<\/p>\n<p>Plot the points in a coordinate plane. Draw a line segment connecting the points.<\/p>\n<p>Question 4.<br \/>\n(1, 1) and (4, 3)<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-24454\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_1-300x229.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 img_1\" width=\"300\" height=\"229\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_1-300x229.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_1.jpg 574w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Question 5.<br \/>\n(2, 3) and (5, 9)<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-24457\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2-300x229.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_2\" width=\"300\" height=\"229\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2-300x229.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2.jpg 574w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Question 6.<br \/>\n(2, 5) and (4, 8)<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-24459\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2-1-300x229.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_2\" width=\"300\" height=\"229\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2-1-300x229.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_2-1.jpg 574w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Use the Distributive Property to \ufb01nd the quotient. Justify your answer.<\/p>\n<p>Question 7.<br \/>\n408 \u00f7 4<\/p>\n<p>Answer: 120<br \/>\nWrite 408 as 204 and 204<br \/>\n204 \u00f7 4 = 51<br \/>\n204 \u00f7 4 = 51<br \/>\n51 + 51 = 102<br \/>\n408 \u00f7 4 = 102<\/p>\n<p>Question 8.<br \/>\n628 \u00f7 2<\/p>\n<p>Answer: 314<br \/>\n608 can be written as 314 and 314<br \/>\n314 \u00f7 2 = 157<br \/>\n314 \u00f7 2 = 157<br \/>\n157 + 157 = 314<br \/>\n628 \u00f7 2 = 314<\/p>\n<p>Question 9.<br \/>\n969 \u00f7 3<\/p>\n<p>Answer: 323<br \/>\n969 can be written as 900 and 69<br \/>\n900 \u00f7 3 = 300<br \/>\n69 \u00f7 3 = 23<br \/>\n300 + 23 = 323<br \/>\n969 \u00f7 3 = 323<\/p>\n<p>Classify the triangle in as many ways as possible.<\/p>\n<p>Question 10.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17031\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-58.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 58\" width=\"118\" height=\"114\" \/><\/p>\n<p>Answer: Acute<br \/>\nAcute angles measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees.<\/p>\n<p>Question 11.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17032\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-59.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 59\" width=\"145\" height=\"127\" \/><\/p>\n<p>Answer: Obtuse<br \/>\nAn obtuse angle has a measurement greater than 90 degrees but less than 180 degrees. However, A reflex angle measures more than 180 degrees but less than 360 degrees.<\/p>\n<p>Question 12.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17033\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-60.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 60\" width=\"125\" height=\"130\" \/><\/p>\n<p>Answer: Right angle<br \/>\nIn geometry and trigonometry, a right angle is an angle of exactly 90\u00b0 (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.<\/p>\n<p><strong>Concepts, Skills, &amp; Problem Solving<\/strong><\/p>\n<p><strong>REWRITING A NUMBER<\/strong><br \/>\n<strong>Write the number as a product of as many factors as possible.<\/strong> (See Exploration 1, p. 15.)<\/p>\n<p>Question 13.<br \/>\n60<\/p>\n<p>Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60<\/p>\n<p>The factors of 60 are:<br \/>\n1 \u00d7 60 = 60<br \/>\n2 \u00d7 30 = 60<br \/>\n3 \u00d7 20 = 60<br \/>\n4 \u00d7 15 = 60<br \/>\n5 \u00d7 12 = 60<br \/>\n6 \u00d7 10 = 60<br \/>\n10 \u00d7 6 = 60<br \/>\n12 \u00d7 5 = 60<br \/>\n15 \u00d7 4 = 60<br \/>\n20 \u00d7 3 = 60<br \/>\n30 \u00d7 2 = 60<br \/>\n60 \u00d7 1 = 60<\/p>\n<p>Question 14.<br \/>\n63<\/p>\n<p>Answer: The factors of 63 are 1, 3, 7, 9, 21, 63<\/p>\n<p>The factors of 63 are:<br \/>\n1 \u00d7 63 = 63<br \/>\n3 \u00d7 21 = 63<br \/>\n7 \u00d7 9 = 63<br \/>\n9 \u00d7 7 = 63<br \/>\n21 \u00d7 3 = 63<br \/>\n63 \u00d7 1 = 63<\/p>\n<p>Question 15.<br \/>\n120<\/p>\n<p>Answer: The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120<\/p>\n<p>The factors of 120 are:<br \/>\n1 \u00d7 120 = 120<br \/>\n2 \u00d7 60 = 120<br \/>\n3 \u00d7 40 = 120<br \/>\n4 \u00d7 30 = 120<br \/>\n5 \u00d7 24 = 120<br \/>\n6 \u00d7 20 = 120<br \/>\n8 \u00d7 15 = 120<br \/>\n10 \u00d7 12 = 120<br \/>\n12 \u00d7 10 = 120<br \/>\n15 \u00d7 8 = 120<br \/>\n20 \u00d7 6 = 120<br \/>\n24 \u00d7 5 = 120<br \/>\n30 \u00d7 4 = 120<br \/>\n40 \u00d7 3 = 120<br \/>\n60 \u00d7 2 = 120<br \/>\n120 \u00d7 1 = 120<\/p>\n<p>Question 16.<br \/>\n150<\/p>\n<p>Answer: The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150<\/p>\n<p>The factors of 150 are:<br \/>\n1 \u00d7 150 = 150<br \/>\n2 \u00d7 75 = 150<br \/>\n3 \u00d7 50 = 150<br \/>\n5 \u00d7 30 = 150<br \/>\n6 \u00d7 25 = 150<br \/>\n10 \u00d7 15 = 150<\/p>\n<p><strong>FINDING FACTOR PAIRS<\/strong><br \/>\n<strong>List the factor pairs of the number.<\/strong><\/p>\n<p>Question 17.<br \/>\n15<\/p>\n<p>Answer: The factor pairs of 15 are (1, 15), (3, 5)<br \/>\n1 \u00d7 15 = 15<br \/>\n3 \u00d7 5 = 15<br \/>\n5 \u00d7 3 = 15<br \/>\n15 \u00d7 1 = 15<\/p>\n<p>Question 18.<br \/>\n22<\/p>\n<p>Answer: The factor pairs of 22 are (1, 22) (2, 11)<br \/>\n1 \u00d7 22 = 22<br \/>\n2 \u00d7 11 = 22<br \/>\n11 \u00d7 2 = 22<br \/>\n22 \u00d7 1 = 22<\/p>\n<p>Question 19.<br \/>\n34<\/p>\n<p>Answer: (1, 34) (2,17)<\/p>\n<p>The factor pairs of 34 are<br \/>\n1 \u00d7 34 = 34<br \/>\n2 \u00d7 17 = 34<br \/>\n17 \u00d7 2 = 34<br \/>\n34 \u00d7 1 = 34<\/p>\n<p>Question 20.<br \/>\n39<\/p>\n<p>Answer: (1, 39) (3, 13)<\/p>\n<p>The factor pairs of 39 are<br \/>\n1 \u00d7 39 = 39<br \/>\n3 \u00d7 13 = 39<br \/>\n13 \u00d7 3 = 39<br \/>\n39 \u00d7 1 = 39<\/p>\n<p>Question 21.<br \/>\n45<\/p>\n<p>Answer: (1, 45) (3, 15) (5, 9)<\/p>\n<p>The factor pairs of 45 are<br \/>\n1 \u00d7 45 = 45<br \/>\n3 \u00d7 15 = 45<br \/>\n5 \u00d7 9 = 45<\/p>\n<p>Question 22.<br \/>\n54<\/p>\n<p>Answer: (1, 54) (2, 27) (3, 18) (6, 9)<\/p>\n<p>The factor pairs of 54 are<br \/>\n1 \u00d7 54 = 54<br \/>\n2 \u00d7 27 = 54<br \/>\n3 \u00d7 18 = 54<br \/>\n6 \u00d7 9 = 54<\/p>\n<p>Question 23.<br \/>\n59<\/p>\n<p>Answer: (1, 59)<br \/>\nThe factor pairs of 59 are<br \/>\n1 \u00d7 59 = 59<br \/>\n59 \u00d7 1 = 59<\/p>\n<p>Question 24.<br \/>\n61<\/p>\n<p>Answer: (1, 61)<br \/>\nThe factor pairs of 61 are<br \/>\n1 \u00d7 61 = 61<br \/>\n61 \u00d7 1 = 61<\/p>\n<p>Question 25.<br \/>\n100<\/p>\n<p>Answer: (1, 100) (2, 50) (4, 25) (5, 20) (10, 10)<br \/>\nThe factor pairs of 100 are<br \/>\n1 \u00d7 100 = 100<br \/>\n2 \u00d7 50 = 100<br \/>\n4 \u00d7 25 = 100<br \/>\n5 \u00d7 20 = 100<br \/>\n10 \u00d7 10 = 100<\/p>\n<p>Question 26.<br \/>\n58<\/p>\n<p>Answer: (1, 58) (2, 29)<br \/>\nThe factor pairs of 58 are<br \/>\n1 \u00d7 58 = 58<br \/>\n2 \u00d7 29 = 58<\/p>\n<p>Question 27.<br \/>\n25<\/p>\n<p>Answer: (1, 25) (5, 5)<br \/>\nThe factor pairs of 25 are<br \/>\n1 \u00d7 25 = 25<br \/>\n5 \u00d7 5 = 25<\/p>\n<p>Question 28.<br \/>\n76<\/p>\n<p>Answer: (1, 76) (2, 38) (4, 19)<br \/>\nThe factor pairs of 76 are<br \/>\n1 \u00d7 76 = 76<br \/>\n2 \u00d7 38 = 76<br \/>\n4 \u00d7 19 = 76<\/p>\n<p>Question 29.<br \/>\n52<\/p>\n<p>Answer: (1, 52) (2, 26) (4, 13)<br \/>\nThe factor pairs of 52 are<br \/>\n1 \u00d7 52 = 52<br \/>\n2 \u00d7 26 = 52<br \/>\n4 \u00d7 13 = 52<\/p>\n<p>Question 30.<br \/>\n88<\/p>\n<p>Answer: (1,88) (2,44) (4, 22) (8, 11)<br \/>\nThe factor pairs of 88 are<br \/>\n1 \u00d7 88 = 88<br \/>\n2 \u00d7 44 = 88<br \/>\n4 \u00d7 22 = 88<br \/>\n8 \u00d7 11 = 88<\/p>\n<p>Question 31.<br \/>\n71<\/p>\n<p>Answer: (1,71)<br \/>\nThe factor pairs of 71 are<br \/>\n1 \u00d7 71 = 71<\/p>\n<p>Question 32.<br \/>\n91<\/p>\n<p>Answer: (1, 91) (7, 13)<br \/>\nThe factor pairs of 91 are<br \/>\n1 \u00d7 91 = 91<br \/>\n7 \u00d7 13 = 91<\/p>\n<p><strong>WRITING A PRIME FACTORIZATION<\/strong><br \/>\n<strong>Write the prime factorization of the number.<\/strong><\/p>\n<p>Question 33.<br \/>\n16<\/p>\n<p>Answer:<br \/>\n16 = 2 \u00d7 8<br \/>\n2 \u00d7 2 \u00d7 4<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 2<\/p>\n<p>Question 34.<br \/>\n25<\/p>\n<p>Answer:<br \/>\n25 = 5 \u00d7 5<\/p>\n<p>Question 35.<br \/>\n30<\/p>\n<p>Answer:<br \/>\n30 = 2 \u00d7 15<br \/>\n= 2 \u00d7 3 \u00d7 5<\/p>\n<p>Question 36.<br \/>\n26<\/p>\n<p>Answer:<br \/>\n26 = 2 \u00d7 13<\/p>\n<p>Question 37.<br \/>\n84<\/p>\n<p>Answer:<br \/>\n84 = 2 \u00d7 42<br \/>\n2 \u00d7 2 \u00d7 21<br \/>\n2 \u00d7 2 \u00d7 3 \u00d7 7<\/p>\n<p>Question 38.<br \/>\n54<\/p>\n<p>Answer:<br \/>\n54 = 2 \u00d7 27<br \/>\n2 \u00d7 3 \u00d7 9<br \/>\n2 \u00d7 3 \u00d7 3 \u00d7 3<\/p>\n<p>Question 39.<br \/>\n65<\/p>\n<p>Answer:<br \/>\n65 = 5 \u00d7 13<\/p>\n<p>Question 40.<br \/>\n77<\/p>\n<p>Answer:<br \/>\n77 = 7 \u00d7 11<\/p>\n<p>Question 41.<br \/>\n46<\/p>\n<p>Answer:<br \/>\n46 = 2 \u00d7 23<\/p>\n<p>Question 42.<br \/>\n39<\/p>\n<p>Answer:<br \/>\n39 = 3 \u00d7 13<\/p>\n<p>Question 43.<br \/>\n99<\/p>\n<p>Answer:<br \/>\n99 = 3 \u00d7 33<br \/>\n3 \u00d7 3 \u00d7 11<\/p>\n<p>Question 44.<br \/>\n24<\/p>\n<p>Answer:<br \/>\n24 = 2 \u00d7 12<br \/>\n2 \u00d7 2 \u00d7 6<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 3<\/p>\n<p>Question 45.<br \/>\n315<\/p>\n<p>Answer:<br \/>\n315 = 3 \u00d7 105<br \/>\n3 \u00d7 3 \u00d7 35<br \/>\n3 \u00d7 3 \u00d7 5 \u00d7 7<\/p>\n<p>Question 46.<br \/>\n490<\/p>\n<p>Answer:<br \/>\n490 = 2 \u00d7 245<br \/>\n2 \u00d7 5 \u00d7 49<br \/>\n2 \u00d7 5 \u00d7 7 \u00d7 7<\/p>\n<p>Question 47.<br \/>\n140<\/p>\n<p>Answer:<br \/>\n2 \u00d7 70<br \/>\n2 \u00d7 2 \u00d7 35<br \/>\n2 \u00d7 2 \u00d7 5 \u00d7 7<\/p>\n<p>Question 48.<br \/>\n640<\/p>\n<p>Answer:<br \/>\n640 = 2 \u00d7 320<br \/>\n2 \u00d7 2 \u00d7 160<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 80<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 40<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 20<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 10<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 5<\/p>\n<p><strong>USING A PRIME FACTORIZATION<\/strong><br \/>\n<strong>Find the number represented by the prime factorization.<\/strong><\/p>\n<p>Question 49.<br \/>\n2<sup>2<\/sup>.3<sup>2<\/sup>.5<\/p>\n<p>Answer:<br \/>\n4 \u00d7 9 \u00d7 5 = 180<br \/>\nWe have to find the prime factorization for 180.<br \/>\n180 = 2 \u00d7 90<br \/>\n2 \u00d7 2 \u00d7 45<br \/>\n2 \u00d7 2\u00d7 3 \u00d7 15<br \/>\n2 \u00d7 2 \u00d7 3 \u00d7 3 \u00d7 5<\/p>\n<p>Question 50.<br \/>\n3<sup>2<\/sup>.5<sup>2<\/sup>.7<\/p>\n<p>Answer:<br \/>\n9 \u00d7 25 \u00d7 7 = 1575<br \/>\nWe have to find the prime factorization for 1575.<br \/>\n1575 = 3 \u00d7 525<br \/>\n3 \u00d7 3 \u00d7 175<br \/>\n3 \u00d7 3 \u00d7 5 \u00d7 35<br \/>\n3 \u00d7 3 \u00d7 5 \u00d7 5 \u00d7 7<\/p>\n<p>Question 51.<br \/>\n2<sup>3<\/sup>.11<sup>2<\/sup>.13<\/p>\n<p>Answer:<br \/>\n8 \u00d7 11 \u00d7 13 = 1144<br \/>\nWe have to find the prime factorization for 1144.<br \/>\n1144 = 2 \u00d7 572<br \/>\n2 \u00d7 2 \u00d7 286<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 143<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 11 \u00d7 13<\/p>\n<p>Question 52.<br \/>\n<strong>YOU BE THE TEACHER<\/strong><br \/>\nYour friend \ufb01nds the prime factorization of 72. Is your friend correct? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17034\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-61.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 61\" width=\"305\" height=\"159\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-61.png 305w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-61-300x156.png 300w\" sizes=\"(max-width: 305px) 100vw, 305px\" \/><\/p>\n<p>Answer:<br \/>\n72 = 2 \u00d7 36<br \/>\n2 \u00d7 2 \u00d7 18<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 9<br \/>\n2 \u00d7 2 \u00d7 2 \u00d7 3 \u00d7 3<br \/>\nYour friend is incorrect because you have to write the prime factorization for 9 also.<br \/>\nThus the prime factorization for 72 is 2 \u00d7 2 \u00d7 2 \u00d7 3 \u00d7 3<\/p>\n<p><strong>USING A PRIME FACTORIZATION<\/strong><br \/>\n<strong>Find the greatest perfect square that is a factor of the number.<\/strong><\/p>\n<p>Question 53.<br \/>\n250<\/p>\n<p>Answer:<br \/>\nA = Calculate the square root of the greatest perfect square from the list of all factors of 250. The factors of 250 are 1, 2, 5, 10, 25, 50, 125, and 250. Furthermore, the greatest perfect square on this list is 25 and the square root of 25 is 5. Therefore, A equals 5.<br \/>\nB = Calculate 250 divided by the greatest perfect square from the list of all factors of 250. We determined above that the greatest perfect square from the list of all factors of 250 is 25. Furthermore, 250 divided by 25 is 10, therefore B equals 10.<br \/>\nNow we have A and B and can get our answer to 250 in its simplest radical form as follows:<br \/>\n\u221a250 = A\u221aB<br \/>\n\u221a250 = 5\u221a10<\/p>\n<p>Question 54.<br \/>\n275<\/p>\n<p>Answer:<br \/>\nA = Calculate the square root of the greatest perfect square from the list of all factors of 275. The factors of 275 are 1, 5, 11, 25, 55, and 275. Furthermore, the greatest perfect square on this list is 25 and the square root of 25 is 5. Therefore, A equals 5.<br \/>\nB = Calculate 275 divided by the greatest perfect square from the list of all factors of 275. We determined above that the greatest perfect square from the list of all factors of 275 is 25. Furthermore, 275 divided by 25 is 11, therefore B equals 11.<br \/>\nNow we have A and B and can get our answer to 275 in its simplest radical form as follows:<br \/>\n\u221a275 = A\u221aB<br \/>\n\u221a275 = 5\u221a11<\/p>\n<p>Question 55.<br \/>\n392<\/p>\n<p>Answer:<br \/>\nA = Calculate the square root of the greatest perfect square from the list of all factors of 392. The factors of 392 are 1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, and 392. Furthermore, the greatest perfect square on this list is 196 and the square root of 196 is 14. Therefore, A equals 14.<br \/>\nB = Calculate 392 divided by the greatest perfect square from the list of all factors of 392. We determined above that the greatest perfect square from the list of all factors of 392 is 196. Furthermore, 392 divided by 196 is 2, therefore B equals 2.<br \/>\nNow we have A and B and can get our answer to 392 in its simplest radical form as follows:<br \/>\n\u221a392 = A\u221aB<br \/>\n\u221a392 = 14\u221a2<\/p>\n<p>Question 56.<br \/>\n338<\/p>\n<p>Answer:<br \/>\nA = Calculate the square root of the greatest perfect square from the list of all factors of 338. The factors of 338 are 1, 2, 13, 26, 169, 338. Furthermore, the greatest perfect square on this list is 324 and the square root of 324 is 18. Therefore, A equals 18.<br \/>\nB = Calculate 338 divided by the greatest perfect square from the list of all factors of 338. We determined above that the greatest perfect square from the list of all factors of 338 is 324.<br \/>\nNow we have A and B and can get our answer to 338 in its simplest radical form as follows:<br \/>\n\u221a338= A\u221aB<br \/>\n\u221a392 = 18\u221a14<\/p>\n<p>Question 57.<br \/>\n244<\/p>\n<p>Answer:<br \/>\nOur first step would be to find out all the factors of 244<br \/>\nSince this is an even number, we divide 2 we get the factors as 244\/2 = 122<br \/>\nNext, we again divide by 2 we get the factor as 122\/2 = 61<br \/>\nWe cannot go down any further since 61 is a prime number<br \/>\nSince we divided by 2, 2 is itself a factor.<br \/>\nLastly, we divided by 2, twice; hence 2*2 = 4 is also a factor<br \/>\nThe factors of 244 are 2,4,61 and 122<br \/>\nOut of 2,4,61 and 122, the only perfect square is 4<br \/>\nSo, the greatest perfect square that is a factor of the number 244 should be 4<br \/>\nTherefore, the answer is: 4<\/p>\n<p>Question 58.<br \/>\n650<\/p>\n<p>Answer:<br \/>\nfactor 650 and find the pairs<br \/>\n650=2\u00d75\u00d75\u00d713<br \/>\nthe pair is 5\u00d75 which is 5\u00b2 or 25<br \/>\nthe greatest perfect square that is a factor of 650 is 25.<\/p>\n<p>Question 59.<br \/>\n756<\/p>\n<p>Answer:<br \/>\nA = Calculate the square root of the greatest perfect square from the list of all factors of 756. The factors of 756 are 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, and 756. Furthermore, the greatest perfect square on this list is 36 and the square root of 36 is 6. Therefore, A equals 6.<br \/>\nB = Calculate 756 divided by the greatest perfect square from the list of all factors of 756. We determined above that the greatest perfect square from the list of all factors of 756 is 36. Furthermore, 756 divided by 36 is 21, therefore B equals 21.<br \/>\nNow we have A and B and can get our answer to 756 in its simplest radical form as follows:<br \/>\n\u221a756 = A\u221aB<br \/>\n\u221a756 = 6\u221a21<\/p>\n<p>Question 60.<br \/>\n1290<\/p>\n<p>Answer:<br \/>\nThere is no greatest perfect square that is a factor of 1290.<br \/>\nThe factors of 1290 are 1, 2, 3, 5, 6, 10, 15, 30, 43, 86, 129, 215, 258, 430, 645, 1290. There are no perfect squares as factors.<\/p>\n<p>Question 61.<br \/>\n2205<\/p>\n<p>Answer: No, the number 2,205 is not a perfect square.<br \/>\nThe factors of 2205 are 1, 3, 5, 7, 9, 15, 21, 35, 45, 49, 63, 105, 147, 245, 315, 441, 735. There are no perfect squares as factors.<\/p>\n<p>Question 62.<br \/>\n1890<\/p>\n<p>Answer: 1890 is not the perfect square.<br \/>\nThe factors of 1890 are 1, 3, 9, 27, 67, 201, 603, 1809. There are no perfect squares as factors.<\/p>\n<p>Question 63.<br \/>\n495<\/p>\n<p>Answer: 495 is not the perfect square.<br \/>\nThe factors 495 are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. There are no perfect squares as factors.<\/p>\n<p>Question 64.<br \/>\n4725<\/p>\n<p>Answer: 4725 is not the perfect square.<br \/>\nThe factors of 4725 are 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 63, 75, 105, 135, 175, 189, 225, 315, 525, 675, 945, 1575, 4725. There are no perfect squares as factors.<\/p>\n<p>Question 65.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nA botanist separates plants into equal groups of 5 for an experiment. Is the total number of plants in the experiment prime or composite? Explain.<\/p>\n<p>Answer: The total number of plants will be a composite.<\/p>\n<p>Explanation:<br \/>\nShe is separating them into equal groups of five. So the total number will be a multiple of 5. Five is a prime number. Multiples of prime numbers are composite numbers.<\/p>\n<p>Question 66.<br \/>\n<strong>REASONING<\/strong><br \/>\nA teacher divides 36 students into equal groups for a scavenger hunt. Each group should have at least 4 students but no more than 8 students. What are the possible group sizes?<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-17035\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-62.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 62\" width=\"207\" height=\"275\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA teacher divides 36 students into equal groups for a scavenger hunt.<br \/>\nEach group should have at least 4 students but no more than 8 students.<br \/>\nThe factors of 36 are 1, 2, 3, 4, 6, 9<br \/>\nSo, there are 6 groups of 6, 9 groups of 4.<\/p>\n<p>Question 67.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nIs 2 the only even prime number? Explain.<\/p>\n<p>Answer:<br \/>\nThe definition of a prime number is a positive integer that has exactly two distinct divisors. Since the divisors of 2 are 1 and 2, there are exactly two distinct divisors, so 2 is prime.<\/p>\n<p>Question 68.<br \/>\n<strong>LOGIC<\/strong><br \/>\nOne table at a bake sale has 75 cookies. Another table has 60 cupcakes. Which table allows for more rectangular arrangements? Explain.<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nOne table at a bake sale has 75 cookies. Another table has 60 cupcakes.<br \/>\n75 = 3\u00b75\u00b2, so has 6 divisors. 6 rectangles are possible if you make the distinction between 1\u00d775 and 75\u00d71.<br \/>\n60 = 2\u00b2\u00b73\u00b75, so has 12 divisors. 12 rectangles are possible under the same conditions.<\/p>\n<p>Question 69.<br \/>\n<strong>PERFECT NUMBERS<\/strong><br \/>\nA perfect number is a number that equals the sum of its factors, not including itself. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28. Because 1 + 2 + 4 + 7 + 14 = 28, 28 is a perfect number. What are the perfect numbers between 1 and 27?<\/p>\n<p>Answer: Perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.<\/p>\n<p>Question 70.<br \/>\n<strong>REPEATED REASONING<\/strong><br \/>\nChoose any two perfect squares and \ufb01nd their product. Then multiply your answer by another perfect square. Continue this process. Are any of the products perfect squares? What can you conclude?<\/p>\n<p>Answer:<br \/>\n2\u00b2 \u00d7 3\u00b2 = 4\u00d79=36, which is a square number<\/p>\n<p>Question 71.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nThe stage manager of a school play creates a rectangular stage that has whole number dimensions and an area of 42 square yards. String lights will outline the stage. What is the least number of yards of string lights needed to enclose the stage?<\/p>\n<p>Answer:<br \/>\nGiven that the stage manager of a school play creates a rectangular acting area of 42 square yards.<br \/>\nLet the length of the rectangular acting area be x, then the width is given by 42 \/ x.<br \/>\nThe number of yards of string lights that the manager needs to enclose the area is given by the perimeter of the rectangular area.<br \/>\nRecall that the perimeter of a rectangle is given by<br \/>\nP = 2(length + width) = 2(x + 42\/x) = 2x + 84\/x<br \/>\nThe perimeter is minimum when the differentiation of 2x + 84\/x is equal to 0.<br \/>\nTherefore, the minimum number of yards of string lights the manager need to enclose in this area is given by<br \/>\n2x &#8211; 84\/x = 0<br \/>\n2x\u00b2 &#8211; 84 = 0<br \/>\n2x\u00b2 = 84<br \/>\nx\u00b2 = 84\/2<br \/>\nx\u00b2 = 42<br \/>\nx \u2248 6.48<\/p>\n<p>Question 72.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nConsider the rectangular prism shown. Using only whole number dimensions, how many different prisms are possible? Explain.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17036\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-63.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 63\" width=\"206\" height=\"155\" \/><\/p>\n<p>Answer:<br \/>\nThe volume of the rectangular prism is lbh<br \/>\nv = 40 cubic inches<br \/>\nLet the length be 5 inches<br \/>\nbreadth be 2 inches<br \/>\nheight be 4 inches<br \/>\nV = 5 \u00d7 2 \u00d7 4<br \/>\nV = 40 cu. inches<\/p>\n<h3><a id=\"Lesson_1.4_Greatest_Common_Factor\"><\/a>Lesson 1.4 Greatest Common Factor<\/h3>\n<p>A Venn diagram uses circles to describe relationships between two or more sets. The Venn diagram shows the factors of 12 and 15. Numbers that are factors of both 12 and 15 are represented by the overlap of the two circles.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17037\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-64.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 64\" width=\"191\" height=\"129\" \/><\/p>\n<p>Answer: 1 and 3 are overlapped between the two circles.<br \/>\nSo, 1 and 3 are the greatest common factors of 12 and 15.<br \/>\nThe factors of 12 are 1, 2, 3, 4, 6, 12<br \/>\nThe factors of 15 are 1, 3, 5, 15.<\/p>\n<p><strong>EXPLORATION 1<\/strong><\/p>\n<p><strong>Identifying Common Factors<\/strong><\/p>\n<p><strong>Work with a partner. In parts (a) \u2013 (d), create a Venn diagram that represents the factors of each number and identify any common factors.<\/strong><br \/>\na. 36 and 48<br \/>\nb. 16 and 56<br \/>\nc. 30 and 75<br \/>\nd. 54 and 90<br \/>\ne. Look at the Venn diagrams in parts (a)\u2013(d). Explain how to identify the greatest common factor of each pair of numbers. Then circle it in each diagram.<\/p>\n<p>Answer:<br \/>\na. <img loading=\"lazy\" class=\"alignnone size-full wp-image-25120\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_3.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_3\" width=\"246\" height=\"218\" \/><\/p>\n<p>b. <img loading=\"lazy\" class=\"alignnone size-medium wp-image-25122\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_4\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>c. <img loading=\"lazy\" class=\"alignnone size-medium wp-image-25123\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_5\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>d. <img loading=\"lazy\" class=\"alignnone size-medium wp-image-25124\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_6\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>e. 36 and 48 have the greatest common factors.<\/p>\n<p><strong>EXPLORATION 2<\/strong><\/p>\n<p><strong>Using Prime Factors<\/strong><br \/>\n<strong>Work with a partner<\/strong><br \/>\na. Each Venn diagram represents the prime factorizations of two numbers. Identify each pair of numbers. Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17038\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-65.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 65\" width=\"517\" height=\"158\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-65.png 517w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-65-300x92.png 300w\" sizes=\"(max-width: 517px) 100vw, 517px\" \/><\/p>\n<p>Answer:<br \/>\ni. Red 2 \u00d7 3 \u00d7 3 = 18<br \/>\nGreen 3 \u00d7 3 \u00d7 3 = 27<br \/>\nGCF = 9<br \/>\nii. Yellow &#8211; 2 \u00d7 2 \u00d7 3 \u00d7 3 \u00d7 5 = 180<br \/>\nPurple &#8211; 5 \u00d7 11 = 55<br \/>\nGCF = 5<br \/>\nb. Create a Venn diagram that represents the prime factorizations of 36 and 48.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-25120\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_3.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_3\" width=\"246\" height=\"218\" \/><br \/>\nc. Repeat part(b) for the remaining number pairs in Exploration 1.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-25122\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_4\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>d. <strong>STRUCTURE<\/strong><br \/>\nMake a conjecture about the relationship between the greatest common factors you found in Exploration 1 and the numbers in the overlaps of the Venn diagrams you just created.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17040\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-66.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 66\" width=\"154\" height=\"157\" \/><\/p>\n<p>Answer:<br \/>\na.<img loading=\"lazy\" class=\"alignnone size-full wp-image-25120\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_3.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_3\" width=\"246\" height=\"218\" \/><br \/>\nThe GCF between the two numbers 36 and 48 are 1,2,3,4,6,12<\/p>\n<p>b.<img loading=\"lazy\" class=\"alignnone size-medium wp-image-25122\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_4\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_4.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nThe GCF between the two numbers 16 and 56 are 1,2,4,8<br \/>\nc. <img loading=\"lazy\" class=\"alignnone size-medium wp-image-25123\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_5\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_5.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nThe GCF between the two numbers 30 and 75 is 15.<br \/>\nd. <img loading=\"lazy\" class=\"alignnone size-medium wp-image-25124\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x202.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors img_6\" width=\"300\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x202.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6.jpg 351w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><strong>1.4 Lesson<\/strong><\/p>\n<p><strong>Try It<\/strong><\/p>\n<p>Find the GCF of the numbers using lists of factors.<\/p>\n<p>Question 1.<br \/>\n8, 36<\/p>\n<p>Answer:<br \/>\nThe factors of 8 are: 1, 2, 4, 8<br \/>\nThe factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36<br \/>\nThen the greatest common factor is 4.<\/p>\n<p>Question 2.<br \/>\n18, 72<\/p>\n<p>Answer:<br \/>\nThe factors of 18 are: 1, 2, 3, 6, 9, 18<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThen the greatest common factor is 18.<\/p>\n<p>Question 3.<br \/>\n14, 28, 49<\/p>\n<p>Answer:<br \/>\nThe factors of 14 are: 1, 2, 7, 14<br \/>\nThe factors of 28 are: 1, 2, 4, 7, 14, 28<br \/>\nThe factors of 49 are: 1, 7, 49<br \/>\nThen the greatest common factor is 7.<\/p>\n<p>Another way to \ufb01nd the GCF of two or more numbers is by using prime factors. The GCF is the product of the common prime factors of the numbers.<\/p>\n<p><strong>Try It<\/strong><br \/>\nFind the GCF of the numbers using prime factorizations.<\/p>\n<p>Question 4.<br \/>\n20,45<\/p>\n<p>Answer:<br \/>\nFind the prime factorization of 20<br \/>\n20 = 2 \u00d7 2 \u00d7 5<br \/>\nFind the prime factorization of 45<br \/>\n45 = 3 \u00d7 3 \u00d7 5<br \/>\nTo find the GCF, multiply all the prime factors common to both numbers:<br \/>\nTherefore, GCF = 5<\/p>\n<p>Question 5.<br \/>\n32,90<\/p>\n<p>Answer:<br \/>\nFind the prime factorization of 32<br \/>\n32 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2<br \/>\nFind the prime factorization of 90<br \/>\n90 = 2 \u00d7 3 \u00d7 3 \u00d7 5<br \/>\nTo find the GCF, multiply all the prime factors common to both numbers:<br \/>\nTherefore, GCF = 2<\/p>\n<p>Question 6.<br \/>\n45,75,120<\/p>\n<p>Answer:<br \/>\n45= 1,3,5,9,15,45<br \/>\n75= 1,3,5,15,25,75<br \/>\n120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120<br \/>\nGCF is 15<\/p>\n<p><strong>Try It<\/strong><\/p>\n<p>Question 7.<br \/>\nWrite a pair of numbers whose greatest common factor is 10.<\/p>\n<p>Answer:<br \/>\nLet\u2019s first find the greatest common factor (GCF) of two whole numbers. The GCF of two numbers is the greatest number that is a factor of both of the numbers. Take the numbers 50 and 30.<br \/>\n50 = 10 \u00d7 5<br \/>\n30 = 10 \u00d7 3<br \/>\nTheir greatest common factor is 10. since 10 is the greatest factor that both numbers have in common.<\/p>\n<p><strong>Self-Assessment for Concepts &amp; Skills<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p><strong>FINDING THE GCF<\/strong><br \/>\n<strong>Find the GCF of the numbers.<\/strong><\/p>\n<p>Question 8.<br \/>\n16, 40<\/p>\n<p>Answer:<br \/>\nThe factors of 16 are: 1, 2, 4, 8, 16<br \/>\nThe factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40<br \/>\nThen the greatest common factor is 8.<\/p>\n<p>Question 9.<br \/>\n35, 63<\/p>\n<p>Answer:<br \/>\nThe factors of 35 are: 1, 5, 7, 35<br \/>\nThe factors of 63 are: 1, 3, 7, 9, 21, 63<br \/>\nThen the greatest common factor is 7.<\/p>\n<p>Question 10.<br \/>\n18, 72, 144<\/p>\n<p>Answer:<br \/>\nThe factors of 18 are: 1, 2, 3, 6, 9, 18<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThe factors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144<br \/>\nThen the greatest common factor is 18.<\/p>\n<p>Question 11.<br \/>\n<strong>MULTIPLE CHOICE<\/strong><br \/>\nWhich number is not a factor of 10? Explain.<br \/>\nA. 1<br \/>\nB. 2<br \/>\nC. 4<br \/>\nD. 5<\/p>\n<p>Answer: 4<\/p>\n<p>Explanation:<br \/>\nFactors of 10 are: 1, 2, 5, 10<br \/>\nThus the correct answer is option C.<\/p>\n<p>Question 12.<br \/>\n<strong>DIFFERENT WORDS, SAME QUESTION<\/strong><br \/>\nWhich is different? Find \u201cboth\u201d answers.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17039\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-66.1.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 66.1\" width=\"463\" height=\"170\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-66.1.png 463w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-66.1-300x110.png 300w\" sizes=\"(max-width: 463px) 100vw, 463px\" \/><\/p>\n<p>Answer:<br \/>\nThe Greatest Common Factor of 24 and 32 is 8<br \/>\nThe Greatest Common Divisor of 24 and 32 is 4<br \/>\nThe Greatest Common Prime Factor of 24 and 32 is 8<br \/>\nThe product of common prime factors of 24 and 32 is 8.<br \/>\nThe Greatest Common Divisor of 24 and 32 are different from others.<\/p>\n<p><strong>Self &#8211; Assessment for Problem Solving<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p>Question 13.<br \/>\nYou use 30 sandwiches and 42 granola bars to make identical picnic baskets. You make the greatest number of picnic baskets with no food left over. How many sandwiches and how many granola bars are in each basket?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17042\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-67.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 67\" width=\"215\" height=\"164\" \/><\/p>\n<p>Answer: 5 sandwiches and 7 granola in each basket<\/p>\n<p>Explanation:<br \/>\nGiven<br \/>\nRepresent Sandwiches with S and Granola with G<br \/>\nS = 30<br \/>\nG = 42<br \/>\nTo do this, we simply need to determine the ratio of S to G<br \/>\nS:G = 30:42<br \/>\nS:G = 5:7<\/p>\n<p>Question 14.<br \/>\nYou \ufb01ll bags with cookies to give to your friends. You bake 45 chocolate chip cookies, 30 peanut butter cookies, and 15 oatmeal cookies. You want identical groups of cookies in each bag with no cookies left over. What is the greatest number of bags you can make?<\/p>\n<p>Answer: 15 bags<\/p>\n<p>Explanation:<br \/>\nGiven,<br \/>\nYou \ufb01ll bags with cookies to give to your friends. You bake 45 chocolate chip cookies, 30 peanut butter cookies, and 15 oatmeal cookies.<br \/>\nYou want identical groups of cookies in each bag with no cookies leftover.<br \/>\n45 chocolate chip cookies<br \/>\n30 peanut butter cookies<br \/>\n15 oatmeal cookies<br \/>\nSo, the GCF is 15.<\/p>\n<h3><a id=\"Greatest_Common_Factor_Practice_1.4\"><\/a>Greatest Common Factor Practice 1.4<\/h3>\n<p><strong>Review &amp; Refresh<\/strong><\/p>\n<p><strong>List the factor pairs of the number.<\/strong><\/p>\n<p>Question 1.<br \/>\n20<\/p>\n<p>Answer: The factor pairs of 20 are (1, 20) (4,5) (2,10)<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 20 = 20<br \/>\n4 \u00d7 5 = 20<br \/>\n2 \u00d7 10 = 20<\/p>\n<p>Question 2.<br \/>\n16<\/p>\n<p>Answer: The factor pairs of 16 are (1,16), (2, 8) (4,4)<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 16 = 16<br \/>\n2 \u00d7 8 = 16<br \/>\n4 \u00d7 4 = 16<\/p>\n<p>Question 3.<br \/>\n56<\/p>\n<p>Answer: The factor pairs of 56 are (1,56) (7,8) (28,2) (14,4)<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 56 = 56<br \/>\n7 \u00d7 8 = 56<br \/>\n28 \u00d7 2 = 56<br \/>\n14 \u00d7 4 = 56<\/p>\n<p>Question 4.<br \/>\n87<\/p>\n<p>Answer: The factor pairs of 87 are (1,87) (3,29)<\/p>\n<p>Explanation:<br \/>\n1 \u00d7 87 = 87<br \/>\n3 \u00d7 29 = 87<\/p>\n<p><strong>Tell whether the statement is always, sometimes, or never true.<\/strong><\/p>\n<p>Question 5.<br \/>\nA rectangle is a rhombus.<\/p>\n<p>Answer: sometimes<\/p>\n<p>No, because all four sides of a rectangle don&#8217;t have to be equal. However, the sets of rectangles and rhombuses do intersect, and their intersection is the set of square all squares are both a rectangle and a rhombus.<\/p>\n<p>Question 6.<br \/>\nA rhombus is a square.<\/p>\n<p>Answer: true<\/p>\n<p>A square is a special case of a rhombus because it has four equal-length sides and goes above and beyond that to also have four right angles. Every square you see will be a rhombus, but not every rhombus you meet will be a square.<\/p>\n<p>Question 7.<br \/>\nA square is a rectangle.<\/p>\n<p>Answer: not always<\/p>\n<p>A square also fits the definition of a rectangle.<\/p>\n<p>Question 8.<br \/>\nA trapezoid is a parallelogram.<\/p>\n<p>Answer: never true<\/p>\n<p>A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. So a parallelogram is also a trapezoid.<\/p>\n<p><strong>Concepts, Skills, &amp; Problem Solving<\/strong><\/p>\n<p><strong>USING A VENN DIAGRAM<\/strong><br \/>\n<strong>Use a Venn diagram to \ufb01nd the greatest common factor of the numbers.<\/strong> (See Exploration 1, p. 21.)<\/p>\n<p>Question 9.<br \/>\n12,30<\/p>\n<p>Answer: 6<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-26353\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-ideas-math-answers-grade-6-chapter-1-img-3.jpg\" alt=\"Big-ideas-math-answers-grade-6-chapter-1-img-3\" width=\"300\" height=\"236\" \/><\/p>\n<p>Question 10.<br \/>\n32,54<\/p>\n<p>Answer: 2<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-26354\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-ideas-math-grade-6-chapter-1-answer-key-img-4-300x187.jpg\" alt=\"Big-ideas-math-grade-6-chapter-1-answer-key-img-4\" width=\"300\" height=\"187\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-ideas-math-grade-6-chapter-1-answer-key-img-4-300x187.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-ideas-math-grade-6-chapter-1-answer-key-img-4.jpg 378w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Question 11.<br \/>\n24,108<\/p>\n<p>Answer: 12<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-medium wp-image-26361\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-6th-Grade-Chapter-1-Answer-Key-Numerical-Expressions-and-Factors-img_5-300x187.jpg\" alt=\"BIM 6th Grade Chapter 1 Answer Key Numerical Expressions and Factors img_5\" width=\"300\" height=\"187\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-6th-Grade-Chapter-1-Answer-Key-Numerical-Expressions-and-Factors-img_5-300x187.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-6th-Grade-Chapter-1-Answer-Key-Numerical-Expressions-and-Factors-img_5.jpg 378w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><strong>FINDING THE GCF<\/strong><br \/>\n<strong>Find the GCF of the numbers using lists of factors.<\/strong><\/p>\n<p>Question 12.<br \/>\n6, 15<\/p>\n<p>Answer: GCF is 3<\/p>\n<p>Explanation:<br \/>\nThe factors of 6 are: 1,2,3,6<br \/>\nThe factors of 15 are: 1,3,5,15<br \/>\nThe common Factors in 6 and 15 is 3.<br \/>\nThus the greatest common factor is 3.<\/p>\n<p>Question 13.<br \/>\n14, 84<\/p>\n<p>Answer: GCF is 14<\/p>\n<p>Explanation:<br \/>\nThe factors of 14 are: 1,2,7,14<br \/>\nThe factors of 84 are: 1,2,3,4,6,7,12,14,21,28,42 84<br \/>\nThe greatest common factor is 14.<\/p>\n<p>Question 14.<br \/>\n45, 108<\/p>\n<p>Answer: GCF is 9<\/p>\n<p>Explanation:<br \/>\nThe factors of 45 are: 1,3,5,9,15,45<br \/>\nThe factors of 108 are: 1,2,3,4,6,9,12,18,27,36,54,108<br \/>\nThe greatest common factor is 9.<\/p>\n<p>Question 15.<br \/>\n39, 65<\/p>\n<p>Answer: GCF is 13<\/p>\n<p>Explanation:<br \/>\nThe factors of 39 are: 1,3,13,39<br \/>\nThe factors of 65 are: 1,5,13,65<br \/>\nThus the greatest common factor is 13.<\/p>\n<p>Question 16.<br \/>\n51, 85<\/p>\n<p>Answer: GCF is 17<\/p>\n<p>Explanation:<br \/>\nThe factors of 51 are: 1,3,17,51<br \/>\nThe factors of 1,5,17,85<br \/>\nThus the greatest common factor is 17<\/p>\n<p>Question 17.<br \/>\n40, 63<\/p>\n<p>Answer: GCF is 1<\/p>\n<p>Explanation:<br \/>\nThe factors of 40 are: 1,2,4,5,8,10,20,40<br \/>\nThe factors of 63 are: 1,3,7,9,21,63<br \/>\nThus the greatest common factor is 1.<\/p>\n<p>Question 18.<br \/>\n12, 48<\/p>\n<p>Answer: GCF is 12<\/p>\n<p>Explanation:<br \/>\nThe factors of 12 are: 1,2,3,4,6,12<br \/>\nThe factors of 48 are: 1,2,3,4,6,8,12,16,24,48<br \/>\nThus the greatest common factor is 12.<\/p>\n<p>Question 19.<br \/>\n24, 52<\/p>\n<p>Answer: GCF is 4<\/p>\n<p>Explanation:<br \/>\nThe factors of 24 are: 1,2,3,4,6,8,12,24<br \/>\nThe factors of 1,2,4,13,36,52<br \/>\nThus the greatest common factor is 4.<\/p>\n<p>Question 20.<br \/>\n30, 58<\/p>\n<p>Answer: GCF is 2<\/p>\n<p>Explanation:<br \/>\nThe factors of 30 are: 1,2,3,5,6,10,15,30<br \/>\nThe factors of 58 are: 1,2,29,58<br \/>\nThus the greatest common factor is 2.<\/p>\n<p><strong>FINDING THE GCF<\/strong><br \/>\n<strong>Find the GCF of the numbers using prime factorizations.<\/strong><\/p>\n<p>Question 21.<br \/>\n45, 60<\/p>\n<p>Answer:<br \/>\nThe prime factorization of 45 is 3 x <strong>3 x 5<\/strong><br \/>\nThe prime factorization 60 is 2 x 2 x <strong>3 x 5<\/strong><br \/>\nGCF of 45, 60 is 3 \u00d7 5 = 15<\/p>\n<p>Question 22.<br \/>\n27, 63<\/p>\n<p>Answer: 9<br \/>\nThe prime factorization of 27 is 3 x <strong>3 x 3<\/strong><br \/>\nThe prime factorization of 63 is<strong> 3 x 3<\/strong> x 7<br \/>\nThus GCF of 27, 63 is 9<\/p>\n<p>Question 23.<br \/>\n36, 81<\/p>\n<p>Answer: 9<br \/>\nThe prime factorization of 36 is 2 x 2 x <strong>3 x 3<\/strong><br \/>\nThe prime factorization of 81 is 3 x 3 x <strong>3 x 3<br \/>\n<\/strong>Thus the GCF of 36, 81 is 9.<\/p>\n<p>Question 24.<br \/>\n72, 84<\/p>\n<p>Answer: 12<br \/>\nThe prime factorization of 72 is 2 x 2 x <strong>2 x 3 x 3<\/strong><br \/>\nThe prime factorization of 84 is <strong>2 x 2 x 3<\/strong> x 7<br \/>\nThus the GCF of 72, 84 is 12.<\/p>\n<p>Question 25.<br \/>\n61, 73<\/p>\n<p>Answer: 1<br \/>\nThe prime factorization of 61 is <strong>1<\/strong> \u00d7 61<br \/>\nThe prime factorization 73 is <strong>1<\/strong> \u00d7 73<br \/>\nThus the GCF of 61, 73 is 1<\/p>\n<p>Question 26.<br \/>\n38, 95<\/p>\n<p>Answer: 19<br \/>\nThe prime factorization of 38 is 2 x <strong>19<\/strong><br \/>\nThe prime factorization of 95 is 5 x <strong>19<\/strong><br \/>\nThus the GCF of 38, 95 is 19<\/p>\n<p>Question 27.<br \/>\n60, 75<\/p>\n<p>Answer: 15<br \/>\nThe prime factorization of 60 is 2 x 2 x <strong>3 x 5<\/strong><br \/>\nThe prime factorization of 75 is <strong>3 x 5<\/strong> x 5<br \/>\nThus the GCF of 60, 75 is 15.<\/p>\n<p>Question 28.<br \/>\n42, 60<\/p>\n<p>Answer: 6<br \/>\nThe prime factorization 42 is <strong>2 \u00d7 3<\/strong> \u00d7 7<br \/>\nThe prime factorization 60 is 2 \u00d7 <strong>2 \u00d7 3<\/strong> \u00d7 5<br \/>\nThus the GCF of 42, 60 is 6<\/p>\n<p>Question 29.<br \/>\n42, 63<\/p>\n<p>Answer: 21<br \/>\nThe prime factorization of 42 is <strong>2 \u00d7 3<\/strong> \u00d7 7<br \/>\nThe prime factorization of 63 is 3 \u00d7<strong> 3 \u00d77<span style=\"font-size: 16px; font-style: normal;\">\u00a0<\/span><\/strong><br \/>\nThus the GCF of 42, 63 is 21<\/p>\n<p>Question 30.<br \/>\n24, 96<\/p>\n<p>Answer: 24<br \/>\nThe prime factorization of 24 is <strong>2 \u00d7 2 \u00d7 2 \u00d7 3<\/strong><br \/>\nThe prime factorization of 96 is 2 x 2 x <strong>2 x 2 x 2 x 3<\/strong><br \/>\nThus the GCF of 24, 96 is 24.<\/p>\n<p>Question 31.<br \/>\n189, 200<\/p>\n<p>Answer: 24<br \/>\nThe prime factorization of 189 is 3 x 3 x 3 x 7<br \/>\nThe prime factorization of 200 is 2 x 2 x 2 x 5 x 5<br \/>\nThus the GCF of 189, 200 is 24.<\/p>\n<p>Question 32.<br \/>\n90, 108<\/p>\n<p>Answer: 18<br \/>\nThe prime factorization of 90 is\u00a0<strong>2 x 3 x 3<\/strong> x 5<br \/>\nThe prime factorization of 108 is 2 x <strong>2 x 3 x 3<\/strong> x 3<br \/>\nThus the GCF of 90, 108 is 18.<\/p>\n<p><strong>OPEN-ENDED<\/strong><br \/>\n<strong>Write a pair of numbers with the indicated GCF.<\/strong><\/p>\n<p>Question 33.<br \/>\n5<\/p>\n<p>Answer: 10, 15<br \/>\nThe factors of 10 are: 1,2,5,10<br \/>\nThe factors of 15 are: 1,3,5,15<br \/>\nThus 10, 15 are pairs of numbers with the indicated GCF 5.<\/p>\n<p>Question 34.<br \/>\n12<\/p>\n<p>Answer: 72, 84<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThe factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84<br \/>\nThen the greatest common factor is 12.<\/p>\n<p>Question 35.<br \/>\n37<\/p>\n<p>Answer: 37,74<br \/>\nThe factors of 37 are: 1,37<br \/>\nThe factors of 74 are: 1,2,37,74<br \/>\nThus 37,74 are pairs of numbers with the indicated GCF 37.<\/p>\n<p>Question 36.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nA teacher is making identical activity packets using 92 crayons and 23 sheets of paper. What is the greatest number of packets the teacher can make with no items left over?<\/p>\n<p>Answer: 23 packets<\/p>\n<p>Explanation:<br \/>\nFactor both numbers:<br \/>\n1. 92=2\u00b746=2\u00b72\u00b723;<br \/>\n2. 23 is a prime number.<br \/>\nThen the greatest common factor GCF (92, 23)=23.<br \/>\nThis means that teacher can make 23 packets, each of them will contain 4 crayons and 1 sheet.<\/p>\n<p>Question 37.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nYou are making balloon arrangements for a birthday party. There are 16 white balloons and 24 red balloons. Each arrangement must be identical. What is the greatest number of arrangements you can make using every balloon?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17041\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-67.1.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 67.1\" width=\"267\" height=\"549\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-67.1.png 267w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-67.1-146x300.png 146w\" sizes=\"(max-width: 267px) 100vw, 267px\" \/><\/p>\n<p>Answer:<br \/>\nThis is a GCF problem. To find the GCF, list the factors.<br \/>\n24<br \/>\n1\u00d724, 2\u00d712, 4\u00d76, 8\u00d73 &#8230;<br \/>\n16<br \/>\n1\u00d716, 2\u00d78, 4\u00d74, &#8230;<br \/>\nThe greatest common factor is 8 because it is the greatest factor of both numbers.<br \/>\nso 8 arrangements, 2 white balloons and 3 red balloons each.<br \/>\nBecause- 8 is the number of groups, and 8\u00d73 =24 so 3 reds, and 2 whites because 8\u00d72=16<\/p>\n<p><strong>YOU BE THE TEACHER<\/strong><br \/>\nYour friend \ufb01nds the GCF of the two numbers. Is your friend correct? Explain your reasoning.<\/p>\n<p>Question 38.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17043\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-68.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 68\" width=\"205\" height=\"121\" \/><\/p>\n<p>Answer:<br \/>\nNo, your friend is incorrect.<br \/>\n42 = <strong>2<\/strong> \u00d7 3 \u00d7 <strong>7<\/strong><br \/>\n154 = <strong>2 \u00d7 7<\/strong> \u00d7 11<br \/>\nThus the GCF is 14.<\/p>\n<p>Question 39.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17044\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-69.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 69\" width=\"217\" height=\"133\" \/><\/p>\n<p>Answer: Yes your friend is correct.<br \/>\nThus the GCF of 36 and 60 is 12.<\/p>\n<p><strong>FINDING THE GCF<\/strong><br \/>\n<strong>Find the GCF of the numbers.<\/strong><\/p>\n<p>Question 40.<br \/>\n35, 56, 63<\/p>\n<p>Answer: GCF is 7<\/p>\n<p>Explanation:<br \/>\nThe factors of 35 are: 1,5,7,35<br \/>\nThe factors of 56 are: 1,2,4,7,8,14,28,56<br \/>\nThe factors of 63 are: 1,3,7,9,21,63<br \/>\nThus the greatest common factor is 7.<\/p>\n<p>Question 41.<br \/>\n30, 60, 78<\/p>\n<p>Answer: GCF is 6<\/p>\n<p>Explanation:<br \/>\nThe factors of 30 are: 1,2,3,5,6,10,15,30<br \/>\nThe factors of 60 are: 1,2,3,4,5,6,10,12,15,20,30,60<br \/>\nThe factors of 78 are: 1,2,3,6,13,26,39,78<br \/>\nThus the greatest common factor is 6.<\/p>\n<p>Question 42.<br \/>\n42, 70, 84<\/p>\n<p>Answer: GCF is 14<\/p>\n<p>The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42<br \/>\nThe factors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70<br \/>\nThe factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84<br \/>\nThen the greatest common factor is 14.<\/p>\n<p>Question 43.<br \/>\n40, 55, 72<\/p>\n<p>Answer: GCF is 1<\/p>\n<p>The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40<br \/>\nThe factors of 55 are: 1, 5, 11, 55<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThen the greatest common factor is 1.<\/p>\n<p>Question 44.<br \/>\n18, 54, 90<\/p>\n<p>Answer: GCF is 18<\/p>\n<p>The factors of 18 are: 1, 2, 3, 6, 9, 18<br \/>\nThe factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54<br \/>\nThe factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90<br \/>\nThen the greatest common factor is 18.<\/p>\n<p>Question 45.<br \/>\n16, 48, 88<\/p>\n<p>Answer: GCF is 8<\/p>\n<p>The factors of 16 are: 1, 2, 4, 8, 16<br \/>\nThe factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48<br \/>\nThe factors of 88 are: 1, 2, 4, 8, 11, 22, 44, 88<br \/>\nThen the greatest common factor is 8.<\/p>\n<p>Question 46.<br \/>\n52, 78, 104<\/p>\n<p>Answer: GCF is 26<\/p>\n<p>The factors of 52 are: 1, 2, 4, 13, 26, 52<br \/>\nThe factors of 78 are: 1, 2, 3, 6, 13, 26, 39, 78<br \/>\nThe factors of 104 are: 1, 2, 4, 8, 13, 26, 52, 104<br \/>\nThen the greatest common factor is 26.<\/p>\n<p>Question 47.<br \/>\n96, 120, 156<\/p>\n<p>Answer: GCF is 12<\/p>\n<p>The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96<br \/>\nThe factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120<br \/>\nThe factors of 156 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156<br \/>\nThen the greatest common factor is 12.<\/p>\n<p>Question 48.<br \/>\n280, 300, 380<\/p>\n<p>Answer: GCF is 20<\/p>\n<p>The factors of 280 are: 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280<br \/>\nThe factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300<br \/>\nThe factors of 380 are: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380<br \/>\nThen the greatest common factor is 20.<\/p>\n<p>Question 49.<br \/>\n<strong>OPEN-ENDED<\/strong><br \/>\nWrite three numbers that have a GCF of 16. What method did you use to \ufb01nd your answer?<\/p>\n<p>Answer: 16, 32, 48<\/p>\n<p>The factors of 16 are: <strong>2 \u00d7 2 \u00d7 2 \u00d7 2<\/strong><br \/>\nThe factors of 32 are : 2 \u00d7 <strong>2 \u00d7 2 \u00d7 2 \u00d7 2<\/strong><br \/>\nThe factors of 48 are: 3 \u00d7<strong> 2 \u00d7 2 \u00d7 2 \u00d7 2<\/strong><\/p>\n<p><strong>CRITICAL THINKING<\/strong><br \/>\n<strong>Tell whether the statement is always, sometimes, or never true. Explain your reasoning.<\/strong><\/p>\n<p>Question 50.<br \/>\nThe GCF of two even numbers is 2.<\/p>\n<p>Answer: Always<\/p>\n<p>Explanation:<br \/>\nExample:<br \/>\nThe factors of 14 are: 1, 2, 7, 14<br \/>\nThe factors of 16 are: 1, 2, 4, 8, 16<br \/>\nThen the greatest common factor is 2.<\/p>\n<p>Question 51.<br \/>\nThe GCF of two prime numbers is 1.<\/p>\n<p>Answer: Always<\/p>\n<p>Explanation:<br \/>\nExample:<br \/>\nThe factors of 3 are: 1, 3<br \/>\nThe factors of 5 are: 1, 5<br \/>\nThen the greatest common factor is 1.<\/p>\n<p>Question 52.<br \/>\nWhen one number is a multiple of another, the GCF of the numbers is the greater of the numbers.<\/p>\n<p>Answer:<br \/>\nWhen one number is a multiple of another, the GCF of the numbers is the greater of the numbers. This is never true since the GCF is a factor of both numbers. So the GCF is the smaller of the two numbers.<\/p>\n<p>Question 53.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nA science museum makes gift bags for students using 168 magnets, 48 robot \ufb01gurines, and 24 packs of freeze-dried ice cream. What is the greatest number of gift bags that can be made using all of the items? How many of each item are in each gift bag?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17045\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-70.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 70\" width=\"236\" height=\"318\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-70.png 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-70-223x300.png 223w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p>Answer:<br \/>\nThe greatest common factor of 24, 48, and 168 is 24, so 24 gift bags can be made. Each will have 1\/24 of the number of gift items of each type that are available.<br \/>\nIn each bag are<br \/>\n1\/24 \u00d7 168 magnets = 7 magnets<br \/>\n1\/24 \u00d7 48 robot figurines = 2 robot figurines<br \/>\n1\/24 \u00d7 24 packs of ice cream = 1 pack of ice cream<\/p>\n<p>Question 54.<br \/>\n<strong>VENN DIAGRAM<\/strong><br \/>\nConsider the numbers 252, 270, and 300.<br \/>\na. Create a Venn diagram using the prime factors of the numbers.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-26932\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x298.jpg\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors img_6\" width=\"300\" height=\"298\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-300x298.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6-150x150.jpg 150w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Expressions-and-Factors-img_6.jpg 314w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nb. Use the Venn diagram to \ufb01nd the GCF of 252, 270, and 300.<\/p>\n<p>Answer:<br \/>\nThe factors of 252 are: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252<br \/>\nThe factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270<br \/>\nThe factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300<br \/>\nThen the greatest common factor is 6.<br \/>\nc. What is the GCF of 252 and 270? 252 and 300? 270 and 300? Explain how you found your answers.<\/p>\n<p>Answer:<br \/>\nThe factors of 252 are: 1, 2, 3, 4, 6, 7, 9, 12, 14, <strong>18<\/strong>, 21, 28, 36, 42, 63, 84, 126, 252<br \/>\nThe factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, <strong>18<\/strong>, 27, 30, 45, 54, 90, 135, 270<br \/>\nThe GCF of 252 and 270 is 18.<br \/>\n252 and 300:<br \/>\nThe factors of 252 are: 1, 2, 3, 4, 6, 7, 9, <strong>12<\/strong>, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252<br \/>\nThe factors of 300 are: 1, 2, 3, 4, 5, 6, 10, <strong>12<\/strong>, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300<br \/>\nThen the greatest common factor is 12.<br \/>\n270 and 300:<br \/>\nThe factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270<br \/>\nThe factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300<br \/>\nThen the greatest common factor is 30.<\/p>\n<p>Question 55.<br \/>\n<strong>REASONING<\/strong><br \/>\nYou are making fruit baskets using 54 apples, 36 oranges, and 73 bananas.<br \/>\na. Explain why you cannot make identical fruit baskets without leftover fruit.<\/p>\n<p>Answer: 73 is a prime number. It can only be divided by 1 and by itself.<br \/>\nb. What is the greatest number of identical fruit baskets you can make with the least amount of fruit left over? Explain how you found your answer.<\/p>\n<p>Answer:<br \/>\nThe GCF of the three numbers:<br \/>\n54 36 73<br \/>\n1\u00d754 1\u00d736 1\u00d773<br \/>\n2\u00d727 2\u00d718<br \/>\n3\u00d718 3\u00d712<br \/>\n6\u00d79 4\u00d79<br \/>\n6\u00d76<br \/>\nGCF of 54, 36, and 73 is 1<br \/>\nGCF of 54 and 36 is 18<br \/>\nIf we divide 54 apples into 18 baskets, we have 3 apples in each basket<br \/>\nIf we divide 36 oranges into 18 baskets, we have 2 oranges in each basket<br \/>\nIf we divide 73 bananas into 18 baskets, we have 4 bananas in each basket + one banana left over.<br \/>\nSo the greatest number of identical fruit baskets we can make with the least amount of fruit left over is 18 baskets<\/p>\n<p>Question 56.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nTwo rectangular, adjacent rooms share a wall. One-foot-by-one-foot tiles cover the \ufb02oor of each room. Describe how the greatest possible length of the adjoining wall is related to the total number of tiles in each room. Draw a diagram that represents one possibility.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17046\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-71.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 71\" width=\"224\" height=\"113\" \/><\/p>\n<p>Answer:<br \/>\nConsider two adjacent rectangular rooms having Length=L, and, Breadth = B<br \/>\nNow Suppose the wall which is in between two rooms has a height or length =H.<br \/>\nThe breadth of wall = B [ if the wall doesn&#8217;t exceed the breadth of the room]<br \/>\nConsidering two rooms to be identical,<br \/>\nArea of each room= L \u00d7 B square unit<br \/>\nArea of each tile = 1\u00d71=1 square unit<br \/>\nNumber of tiles required= L B \u00f7 1= LB tiles( product of length and breadth of the room is the number of tiles required)<br \/>\nSuppose if,LB= N<br \/>\nB= N\/LArea of the wall(W) = B\u00d7H= B H square unit<br \/>\nB =W\/H<br \/>\nEquating (1) and (2)<br \/>\nN\/L = W\/ H<br \/>\nH = WL\/N<br \/>\nH = WL\/LB<br \/>\nH = W\/B<br \/>\nH = Area of wall\/Breadth of room or wall<\/p>\n<h3><a id=\"Lesson_1.5_Least_Common_Multiple\"><\/a>Lesson 1.5 Least Common Multiple<\/h3>\n<p><strong>EXPLORATION 1<br \/>\nIdentifying Common Multiples<\/strong><\/p>\n<p><strong>Work with a partner. In parts (a)\u2013(d), create a Venn diagram that represents the \ufb01rst several multiples of each number and identify any common multiples.<\/strong><br \/>\na. 8 and 12<br \/>\nb. 4 and 14<br \/>\nc. 10 and 15<br \/>\nd. 20 and 35<br \/>\ne. Look at the Venn diagrams in parts (a)\u2013(d). Explain how to identify the least common multiple of each pair of numbers. Then circle it in each diagram.<\/p>\n<p>Answer:<br \/>\na. <img loading=\"lazy\" class=\"alignnone size-full wp-image-27010\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7.jpg\" alt=\"BIM Grade 6 Chapter 1 Answer Key img_7\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><br \/>\nb. <img loading=\"lazy\" class=\"alignnone size-full wp-image-27013\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8.jpg\" alt=\"Big Ideas math 6th grade answers chapter 1 img_8\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><br \/>\nc.<img loading=\"lazy\" class=\"alignnone size-full wp-image-27016\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_9.jpg\" alt=\"Big Ideas math 6th grade answers chapter 1 img_9\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_9.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_9-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p>d. <img loading=\"lazy\" class=\"alignnone size-full wp-image-27019\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_10.png\" alt=\"Big Ideas math 6th grade answers chapter 1 img_10\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_10.png 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_10-150x150.png 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p><strong>EXPLORATION 2<\/strong><br \/>\n<strong>Using Prime Factors<\/strong><br \/>\n<strong>Work with a partner.<\/strong><br \/>\na. Create a Venn diagram that represents the prime factorizations of 8 and 12.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17047\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-72.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 72\" width=\"155\" height=\"143\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-27010\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7.jpg\" alt=\"BIM Grade 6 Chapter 1 Answer Key img_7\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/BIM-Grade-6-Chapter-1-Answer-Key-img_7-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><br \/>\nb. Repeat part (a) for the remaining number pairs in Exploration 1.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-27013\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8.jpg\" alt=\"Big Ideas math 6th grade answers chapter 1 img_8\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-math-6th-grade-answers-chapter-1-img_8-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><br \/>\nc.<strong> STRUCTURE<\/strong><br \/>\nMake a conjecture about the relationship between the least common multiples you found in Exploration 1 and the numbers in the Venn diagrams you just created.<\/p>\n<p>Answer:<br \/>\nThe numbers which are overlapped are the least common multiples of the numbers.<br \/>\nd. The Venn diagram shows the prime factors of two numbers.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17048\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-73.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 73\" width=\"218\" height=\"148\" \/><br \/>\nUse the diagram to complete the following tasks.<\/p>\n<ul>\n<li>Identify the two numbers.<\/li>\n<li>Find the greatest common factor.<\/li>\n<li>Find the least common multiple<\/li>\n<\/ul>\n<p>Answer:<\/p>\n<ul>\n<li>120, 180<\/li>\n<li>The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120The factors of 180 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180Then the greatest common factor is 60.<\/li>\n<li>Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 120:<br \/>\n120, 240, 360, 480, 600<br \/>\nMultiples of 180:<br \/>\n180, 360, 540, 720<br \/>\nTherefore,<br \/>\nLCM(120, 180) = 360<\/li>\n<\/ul>\n<p><strong>1.5 Lesson<\/strong><\/p>\n<p><strong>Try It<\/strong><\/p>\n<p>Find the LCM of the numbers using lists of multiples.<\/p>\n<p>Question 1.<br \/>\n3, 8<\/p>\n<p>Answer: 24<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 3:<br \/>\n3, 6, 9, 12, 15, 18, 21, 24, 27, 30<br \/>\nMultiples of 8:<br \/>\n8, 16, 24, 32, 40<br \/>\nTherefore,<br \/>\nLCM(3, 8) = 24<\/p>\n<p>Question 2.<br \/>\n9, 12<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 9:<br \/>\n9, 18, 27, 36, 45, 54<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60<br \/>\nTherefore,<br \/>\nLCM(9, 12) = 36<\/p>\n<p>Question 3.<br \/>\n6, 10<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 6:<br \/>\n6, 12, 18, 24, 30, 36, 42<br \/>\nMultiples of 10:<br \/>\n10, 20, 30, 40, 50<br \/>\nTherefore,<br \/>\nLCM(6, 10) = 30<\/p>\n<p><strong>Try It<\/strong><br \/>\nFind the LCM of the numbers using prime factorizations.<\/p>\n<p>Question 4.<br \/>\n14, 18<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 14 is:<br \/>\n2 x 7<br \/>\nPrime Factorization of 18 is:<br \/>\n2 x 3 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 3, 3, 7<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 3 x 3 x 7 = 126<br \/>\nLCM(14, 18) = 126<\/p>\n<p>Question 5.<br \/>\n28, 36<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 28 is:<br \/>\n2 x 2 x 7<br \/>\nPrime Factorization of 36 is:<br \/>\n2 x 2 x 3 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 3, 3, 7<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 3 x 3 x 7 = 252<br \/>\nLCM = 252<br \/>\nTherefore,<br \/>\nLCM(28, 36) = 252<\/p>\n<p>Question 6.<br \/>\n24, 90<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 24 is:<br \/>\n2 x 2 x 2 x 3<br \/>\nPrime Factorization of 90 is:<br \/>\n2 x 3 x 3 x 5<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 2, 3, 3, 5<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 2 x 3 x 3 x 5 = 360<br \/>\nLCM = 360<br \/>\nTherefore,<br \/>\nLCM(24, 90) = 360<\/p>\n<p><strong>Try It<\/strong><\/p>\n<p>Find the LCM of the numbers.<\/p>\n<p>Question 7.<br \/>\n2, 5, 8<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 2:<br \/>\n2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44<br \/>\nMultiples of 5:<br \/>\n5, 10, 15, 20, 25, 30, 35, 40, 45, 50<br \/>\nMultiples of 8:<br \/>\n8, 16, 24, 32, 40, 48, 56<br \/>\nTherefore,<br \/>\nLCM(2, 5, 8) = 40<\/p>\n<p>Question 8.<br \/>\n6, 10, 12<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 6:<br \/>\n6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72<br \/>\nMultiples of 10:<br \/>\n10, 20, 30, 40, 50, 60, 70, 80<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60, 72, 84<br \/>\nTherefore,<br \/>\nLCM(6, 10, 12) = 60<\/p>\n<p>Question 9.<br \/>\nWrite three numbers that have a least common multiple of 100.<\/p>\n<p>Answer: 1, 10, 100<\/p>\n<p>Explanation:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 1:<br \/>\n1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, <strong>100<\/strong>, 101, 102<br \/>\nMultiples of 10:<br \/>\n10, 20, 30, 40, 50, 60, 70, 80, 90, <strong>100<\/strong>, 110, 120<br \/>\nMultiples of 100:<br \/>\n<strong>100<\/strong>, 200, 300<br \/>\nTherefore,<br \/>\nLCM(1, 10, 100) = 100<\/p>\n<p><strong>Self-Assessment for Concepts &amp; Skills<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p><strong>FINDING THE LCM<\/strong><br \/>\n<strong>Find the LCM of the numbers.<\/strong><\/p>\n<p>Question 10.<br \/>\n6, 9<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 6:<br \/>\n6, 12, 18, 24, 30<br \/>\nMultiples of 9:<br \/>\n9, 18, 27, 36<br \/>\nTherefore,<br \/>\nLCM(6, 9) = 18<\/p>\n<p>Question 11.<br \/>\n30, 40<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 30:<br \/>\n30, 60, 90, 120, 150, 180<br \/>\nMultiples of 40:<br \/>\n40, 80, 120, 160, 200<br \/>\nTherefore,<br \/>\nLCM(30, 40) = 120<\/p>\n<p>Question 12.<br \/>\n5, 11<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 5:<br \/>\n5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65<br \/>\nMultiples of 11:<br \/>\n11, 22, 33, 44, 55, 66, 77<br \/>\nTherefore,<br \/>\nLCM(5, 11) = 55<\/p>\n<p>Question 13.<br \/>\n<strong>Reasoning<\/strong><br \/>\nWrite two numbers such that 18 and 30 are multiples of the numbers. Justify your answer.<\/p>\n<p>Answer: 3, 6<\/p>\n<p>Explanation:<br \/>\nMultiples of 3: 3, 6, 9, 12, 15, <strong>18<\/strong>, 21, 24, 27, <strong>30<\/strong><br \/>\nMultiples of 6: 6,12,<strong>18<\/strong>,24,<strong>30<\/strong>,36,42,48<br \/>\nThus 18 and 30 are the multiples of 3 and 6.<\/p>\n<p>Question 14.<br \/>\n<strong>REASONING<\/strong><br \/>\nYou need to \ufb01nd the LCM of 13 and 14. Would you rather list their multiples or use their prime factorizations? Explain.<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 13 shows:<br \/>\n13 is prime = 13<br \/>\nPrime Factorization of 14 is:<br \/>\n2 x 7<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 7, 13<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 7 x 13 = 182<br \/>\nIn exponential form:<br \/>\nLCM = 2 x 7 x 13 = 182<br \/>\nLCM = 182<br \/>\nTherefore,<br \/>\nLCM(13, 14) = 182<\/p>\n<p>Question 15.<br \/>\n<strong>CHOOSE TOOLS<\/strong><br \/>\nA student writes the prime factorizations of 8 and 12 in a table as shown. She claims she can use the table to \ufb01nd the greatest common factor and the least common multiple of 8 and 12. How is this possible?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16983\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 74\" width=\"380\" height=\"88\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74.png 380w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74-300x69.png 300w\" sizes=\"(max-width: 380px) 100vw, 380px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16985\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-75.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 75\" width=\"218\" height=\"189\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-medium wp-image-26936\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74-300x69.jpg\" alt=\"Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74\" width=\"300\" height=\"69\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74-300x69.jpg 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-74.jpg 380w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 2, 3<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 2 x 3 = 24<br \/>\nThe least common multiple of 8 and 12 is 24.<\/p>\n<p><strong>Self-Assessment for Problem Solving<\/strong><\/p>\n<p>Solve each exercise. Then rate your understanding of the success criteria in your journal.<\/p>\n<p>Question 17.<br \/>\nA geyser erupts every fourth day. Another geyser erupts every sixth day. Today both geysers erupted. In how many days will both geysers erupt on the same day again?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA geyser erupts every fourth day. Another geyser erupts every sixth day. Today both geysers erupted.<br \/>\nThe geyser erupts on the same day after 12 days.<\/p>\n<p>Question 18.<br \/>\nA water park has two large buckets that slowly \ufb01ll with water. One bucket dumps water every 12 minutes. The other bucket dumps water every 10 minutes. Five minutes ago, both buckets dumped water. When will both buckets dump water at the same time again?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16986\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-76.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 76\" width=\"193\" height=\"272\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA water park has two large buckets that slowly \ufb01ll with water.<br \/>\nOne bucket dumps water every 12 minutes.<br \/>\nThe other bucket dumps water every 10 minutes.<br \/>\nFive minutes ago, both buckets dumped water.<br \/>\nBoth buckets will dump again at the same time in 60 minutes (1 hour.)<\/p>\n<p>Question 19.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nYou purchase disposable plates, cups, and forks for a cookout. Plates are sold in packages of 24, cups in packages of 32, and forks in packages of 48. What are the least numbers of packages you should buy in order to have the same number of plates, cups, and forks?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nYou purchase disposable plates, cups, and forks for a cookout. Plates are sold in packages of 24, cups in packages of 32, and forks in packages of 48.<br \/>\nWe solve this question using the Lowest Common Multiple (LCM) method\/<br \/>\nStep 1<br \/>\nWe list multiples of each number until the first common multiple is found. This is referred to as the lowest common multiple.<br \/>\nPlates are sold in packages of 24<br \/>\nCups in packages of 32<br \/>\nForks in packages of 48<br \/>\nMultiples of 24:<br \/>\n24, 48, 72, 96, 120, 144<br \/>\nMultiples of 32:<br \/>\n32, 64, 96, 128, 160<br \/>\nMultiples of 48:<br \/>\n48, 96, 144, 192<br \/>\nTherefore,<br \/>\nLCM(24, 32, 48) = 96<br \/>\nHence, the least numbers of packages you should buy in order to have the same number of plates, cups, and forks is 96 packages<\/p>\n<h3><a id=\"Least_Common_Multiple_Practice_1.5\"><\/a>Least Common Multiple Practice 1.5<\/h3>\n<p><strong>Review &amp; Refresh<\/strong><\/p>\n<p><strong>Find the GCF of the numbers.<\/strong><\/p>\n<p>Question 1.<br \/>\n18, 42<\/p>\n<p>Answer:<br \/>\nThe factors of 18 are: 1, 2, 3, 6, 9, 18<br \/>\nThe factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42<br \/>\nThen the greatest common factor is 6.<\/p>\n<p>Question 2.<br \/>\n72, 96<\/p>\n<p>Answer:<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThe factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96<br \/>\nThen the greatest common factor is 24.<\/p>\n<p>Question 3.<br \/>\n38, 76, 114<\/p>\n<p>Answer:<br \/>\nThe factors of 38 are: 1, 2, 19, 38<br \/>\nThe factors of 76 are: 1, 2, 4, 19, 38, 76<br \/>\nThe factors of 114 are: 1, 2, 3, 6, 19, 38, 57, 114<br \/>\nThen the greatest common factor is 38.<\/p>\n<p><strong>Divide.<\/strong><\/p>\n<p>Question 4.<br \/>\n900 \u00f7 6<\/p>\n<p>Answer: 150<\/p>\n<p>Explanation:<br \/>\nDivide the two numbers 900 and 6.<br \/>\n900\/6 = 150<br \/>\nIt means 6 divides 900 150 times.<br \/>\nThus the quotient is 150.<\/p>\n<p>Question 5.<br \/>\n1944 \u00f7 9<\/p>\n<p>Answer: 216<\/p>\n<p>Explanation:<br \/>\nDivide the two numbers 1944 and 9<br \/>\n1944\/9 = 216<br \/>\nIt means that 9 divides 1944 216 times.<br \/>\nThus the quotient is 216<\/p>\n<p>Question 6.<br \/>\n672 \u00f7 12<\/p>\n<p>Answer: 56<\/p>\n<p>Explanation:<br \/>\nDivide the two numbers 672 and 12.<br \/>\n672\/12 = 56<br \/>\nIt means 12 divides 672 56 times.<br \/>\nThus the quotient is 56.<\/p>\n<p><strong>Write an ordered pair that corresponds to the point.<\/strong><\/p>\n<p>Question 7.<br \/>\nPoint A<\/p>\n<p>Answer: (2,4)<br \/>\nBy seeing the below graph we can find the ordered pairs.<br \/>\n2 lies on the x-axis and 4 lies on the y-axis.<br \/>\nSo, the ordered pairs to the point A is (2,4)<\/p>\n<p>Question 8.<br \/>\nPoint B<\/p>\n<p>Answer: (3,1)<br \/>\nBy seeing the below graph we can find the ordered pairs.<br \/>\n3 lies on the x-axis and 1 lies on the y-axis.<br \/>\nSo, the ordered pairs to the point A is (3,1)<\/p>\n<p>Question 9.<br \/>\nPoint C<\/p>\n<p>Answer: (4,7)<br \/>\nBy seeing the below graph we can find the ordered pairs.<br \/>\n4 lies on the x-axis and 7 lies on the y-axis.<br \/>\nSo, the ordered pairs to the point A is (4,7)<\/p>\n<p>Question 10.<br \/>\nPoint D<\/p>\n<p>Answer: (9,6)<br \/>\nBy seeing the below graph we can find the ordered pairs.<br \/>\n9 lies on the x-axis and 6 lies on the y-axis.<br \/>\nSo, the ordered pairs to the point A is (9,6)<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-16987\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-77.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 77\" width=\"210\" height=\"207\" \/><\/p>\n<p><strong>Concepts, Skills, &amp; Problem Solving<\/strong><\/p>\n<p><strong>USING A VENN DIAGRAM<\/strong><br \/>\n<strong>Use a Venn diagram to \ufb01nd the least common multiple of the numbers.<\/strong> (See Exploration 1, p. 27.)<\/p>\n<p>Question 11.<br \/>\n3, 7<\/p>\n<p>Answer: 21<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-27759\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_10.jpg\" alt=\"Big Ideas Math Grade 6 Chapter 1 img_10\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_10.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Grade-6-Chapter-1-img_10-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p>Question 12.<br \/>\n6, 8<\/p>\n<p>Answer: 24<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-27760\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_11.jpg\" alt=\"Big Ideas Math 6th Grade Answers Chapter 1 img_11\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_11.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_11-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p>Question 13.<br \/>\n4, 5<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-27761\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_12.jpg\" alt=\"Big Ideas Math 6th Grade Answers Chapter 1 img_12\" width=\"236\" height=\"236\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_12.jpg 236w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-6th-Grade-Answers-Chapter-1-img_12-150x150.jpg 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p><strong>FINDING THE LCM<\/strong><br \/>\n<strong>Find the LCM of the numbers using lists of multiples.<\/strong><\/p>\n<p>Question 14.<br \/>\n1, 5<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 1:<br \/>\n1, 2, 3, 4, 5, 6, 7<br \/>\nMultiples of 5:<br \/>\n5, 10, 15<br \/>\nTherefore,<br \/>\nLCM(1, 5) = 5<\/p>\n<p>Question 15.<br \/>\n2, 6<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 2:<br \/>\n2, 4, 6, 8, 10<br \/>\nMultiples of 6:<br \/>\n6, 12, 18<br \/>\nTherefore,<br \/>\nLCM(2, 6) = 6<\/p>\n<p>Question 16.<br \/>\n2, 3<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 2:<br \/>\n2, 4, 6, 8, 10<br \/>\nMultiples of 3:<br \/>\n3, 6, 9, 12<br \/>\nTherefore,<br \/>\nLCM(2, 3) = 6<\/p>\n<p>Question 17.<br \/>\n2, 9<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 2:<br \/>\n2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22<br \/>\nMultiples of 9:<br \/>\n9, 18, 27, 36<br \/>\nTherefore,<br \/>\nLCM(2, 9) = 18<\/p>\n<p>Question 18.<br \/>\n3, 4<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 3:<br \/>\n3, 6, 9, 12, 15, 18<br \/>\nMultiples of 4:<br \/>\n4, 8, 12, 16, 20<br \/>\nTherefore,<br \/>\nLCM(3, 4) = 12<\/p>\n<p>Question 19.<br \/>\n8, 9<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 8:<br \/>\n8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88<br \/>\nMultiples of 9:<br \/>\n9, 18, 27, 36, 45, 54, 63, 72, 81, 90<br \/>\nTherefore,<br \/>\nLCM(8, 9) = 72<\/p>\n<p>Question 20.<br \/>\n5, 8<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 5:<br \/>\n5, 10, 15, 20, 25, 30, 35, 40, 45, 50<br \/>\nMultiples of 8:<br \/>\n8, 16, 24, 32, 40, 48, 56<br \/>\nTherefore,<br \/>\nLCM(5, 8) = 40<\/p>\n<p>Question 21.<br \/>\n11, 12<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 11:<br \/>\n11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156<br \/>\nTherefore,<br \/>\nLCM(11, 12) = 132<\/p>\n<p>Question 22.<br \/>\n12, 18<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60<br \/>\nMultiples of 18:<br \/>\n18, 36, 54, 72<br \/>\nTherefore,<br \/>\nLCM(12, 18) = 36<\/p>\n<p><strong>FINDING THE LCM<\/strong><br \/>\n<strong>Find the LCM of the numbers using prime factorizations.<\/strong><\/p>\n<p>Question 23.<br \/>\n7, 12<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 7 shows:<br \/>\n7 is prime<br \/>\nPrime Factorization of 12 is:<br \/>\n2 x 2 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 3, 7<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 3 x 7 = 84<br \/>\nLCM = 84<br \/>\nTherefore,<br \/>\nLCM(7, 12) = 84<\/p>\n<p>Question 24.<br \/>\n5, 9 4<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 4 is:<br \/>\n2 x 2<br \/>\nPrime Factorization of 5 shows:<br \/>\n5 is prime<br \/>\nPrime Factorization of 9 is:<br \/>\n3 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 3, 3, 5<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 3 x 3 x 5 = 180<br \/>\nLCM = 180<br \/>\nTherefore,<br \/>\nLCM(4, 5, 9) = 180<\/p>\n<p>Question 25.<br \/>\n4, 11<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 4 is:<br \/>\n2 x 2<br \/>\nPrime Factorization of 11 shows:<br \/>\n11 is prime<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 11<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 11 = 44<br \/>\nLCM = 44<br \/>\nTherefore,<br \/>\nLCM(4, 11) = 44<\/p>\n<p>Question 26.<br \/>\n9, 10<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 9 is:<br \/>\n3 x 3<br \/>\nPrime Factorization of 10 is:<br \/>\n2 x 5<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 3, 3, 5<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 3 x 3 x 5 = 90<br \/>\nLCM = 90<br \/>\nTherefore,<br \/>\nLCM(9, 10) = 90<\/p>\n<p>Question 27.<br \/>\n12, 27<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 12 is:<br \/>\n2 x 2 x 3<br \/>\nPrime Factorization of 27 is:<br \/>\n3 x 3 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 3, 3, 3<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 3 x 3 x 3 = 108<br \/>\nLCM = 108<br \/>\nTherefore,<br \/>\nLCM(12, 27) = 108<\/p>\n<p>Question 28.<br \/>\n18, 45<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 18 is:<br \/>\n2 x 3 x 3<br \/>\nPrime Factorization of 45 is:<br \/>\n3 x 3 x 5<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 3, 3, 5<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 3 x 3 x 5 = 90<br \/>\nLCM = 90<br \/>\nTherefore,<br \/>\nLCM(18, 45) = 90<\/p>\n<p>Question 29.<br \/>\n22, 23<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 22 is:<br \/>\n2 x 11<br \/>\nPrime Factorization of 23 shows:<br \/>\n23 is prime<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 11, 23<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 11 x 23 = 506<br \/>\nLCM = 506<br \/>\nTherefore,<br \/>\nLCM(22, 23) = 506<\/p>\n<p>Question 30.<br \/>\n36, 60<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 36 is:<br \/>\n2 x 2 x 3 x 3<br \/>\nPrime Factorization of 60 is:<br \/>\n2 x 2 x 3 x 5<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 2, 3, 3, 5<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 2 x 3 x 3 x 5 = 180<br \/>\nLCM = 180<br \/>\nTherefore,<br \/>\nLCM(36, 60) = 180<\/p>\n<p>Question 31.<br \/>\n35, 50<\/p>\n<p>Answer:<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 35 is:<br \/>\n5 x 7<br \/>\nPrime Factorization of 50 is:<br \/>\n2 x 5 x 5<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 5, 5, 7<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 5 x 5 x 7 = 350<br \/>\nLCM = 350<br \/>\nTherefore,<br \/>\nLCM(35, 50) = 350<\/p>\n<p>Question 32.<br \/>\n<strong>YOU BE THE TEACHER<\/strong><br \/>\nYour friend \ufb01nds the LCM of 6 and 9. Is your friend correct? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16988\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-78.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 78\" width=\"261\" height=\"111\" \/><\/p>\n<p>Answer: No friend is incorrect.<br \/>\nList all prime factors for each number.<br \/>\nPrime Factorization of 6 is:<br \/>\n2 x 3<br \/>\nPrime Factorization of 9 is:<br \/>\n3 x 3<br \/>\nFor each prime factor, find where it occurs most often as a factor and write it that many times in a new list.<br \/>\nThe new superset list is<br \/>\n2, 3, 3<br \/>\nMultiply these factors together to find the LCM.<br \/>\nLCM = 2 x 3 x 3 = 18<br \/>\nLCM = 18<br \/>\nTherefore,<br \/>\nLCM(6, 9) = 18<\/p>\n<p>Question 33.<br \/>\n<strong>MODELING REAL LIFE<\/strong><br \/>\nYou have diving lessons every \ufb01fth day and swimming lessons every third day. Today you have both lessons. In how many days will you have both lessons on the same day again?<\/p>\n<p>Answer:<br \/>\nAfter 15 days<br \/>\ndiving is on 0 , 5 , 10 , 15 , 20 , &#8230; days<br \/>\nswimming is on 0 , 3, 6 , 9, 12 , 15 , 18 , &#8230; days<br \/>\nThus the next will be in 15 days<\/p>\n<p>Question 34.<br \/>\n<strong>REASONING<\/strong><br \/>\nWhich model represents an LCM that is different from the other three? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17049\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-79.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 79\" width=\"630\" height=\"167\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-79.png 630w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-79-300x80.png 300w\" sizes=\"(max-width: 630px) 100vw, 630px\" \/><\/p>\n<p>Answer: The number line B is different from the other three number lines.<\/p>\n<p><strong>FINDING THE LCM<\/strong><br \/>\n<strong>Find the LCM of the numbers.<\/strong><\/p>\n<p>Question 35.<br \/>\n2, 3, 7<\/p>\n<p>Answer:<br \/>\nLCM(2,3) = (2 \u00d7 3) \/ GCF(2,3)<br \/>\n= (2 \u00d7 3) \/ 1<br \/>\n= 6 \/ 1<br \/>\n= 6<br \/>\nLCM(6,7) = (6 \u00d7 7) \/ GCF(6,7)<br \/>\n= (6 \u00d7 7) \/ 1<br \/>\n= 42 \/ 1<br \/>\n= 42<br \/>\nTherefore,<br \/>\nLCM(2, 3, 7) = 42<\/p>\n<p>Question 36.<br \/>\n3, 5, 11<\/p>\n<p>Answer:<br \/>\nLCM(3,5) = (3 \u00d7 5) \/ GCF(3,5)<br \/>\n= (3 \u00d7 5) \/ 1<br \/>\n= 15 \/ 1<br \/>\n= 15<br \/>\nLCM(15,11) = (15 \u00d7 11) \/ GCF(15,11)<br \/>\n= (15 \u00d7 11) \/ 1<br \/>\n= 165 \/ 1<br \/>\n= 165<br \/>\nTherefore,<br \/>\nLCM(3, 5, 11) = 165<\/p>\n<p>Question 37.<br \/>\n4, 9, 12<\/p>\n<p>Answer:<br \/>\nLCM(4,9) = (4 \u00d7 9) \/ GCF(4,9)<br \/>\n= (4 \u00d7 9) \/ 1<br \/>\n= 36 \/ 1<br \/>\n= 36<br \/>\nLCM(36,12) = (36 \u00d7 12) \/ GCF(36,12)<br \/>\n= (36 \u00d7 12) \/ 12<br \/>\n= 432 \/ 12<br \/>\n= 36<br \/>\nTherefore,<br \/>\nLCM(4, 9, 12) = 36<\/p>\n<p>Question 38.<br \/>\n6, 8, 15<\/p>\n<p>Answer:<br \/>\nLCM(6,8) = (6 \u00d7 8) \/ GCF(6,8)<br \/>\n= (6 \u00d7 8) \/ 2<br \/>\n= 48 \/ 2<br \/>\n= 24<br \/>\nLCM(24,15) = (24 \u00d7 15) \/ GCF(24,15)<br \/>\n= (24 \u00d7 15) \/ 3<br \/>\n= 360 \/ 3<br \/>\n= 120<br \/>\nTherefore,<br \/>\nLCM(6, 8, 15) = 120<\/p>\n<p>Question 39.<br \/>\n7, 18, 21<\/p>\n<p>Answer:<br \/>\nLCM(7,18) = (7 \u00d7 18) \/ GCF(7,18)<br \/>\n= (7 \u00d7 18) \/ 1<br \/>\n= 126 \/ 1<br \/>\n= 126<br \/>\nLCM(126,21) = (126 \u00d7 21) \/ GCF(126,21)<br \/>\n= (126 \u00d7 21) \/ 21<br \/>\n= 2646 \/ 21<br \/>\n= 126<br \/>\nTherefore,<br \/>\nLCM(7, 18, 21) = 126<\/p>\n<p>Question 40.<br \/>\n9, 10, 28<\/p>\n<p>Answer:<br \/>\nLCM(9,10) = (9 \u00d7 10) \/ GCF(9,10)<br \/>\n= (9 \u00d7 10) \/ 1<br \/>\n= 90 \/ 1<br \/>\n= 90<br \/>\nLCM(90,28) = (90 \u00d7 28) \/ GCF(90,28)<br \/>\n= (90 \u00d7 28) \/ 2<br \/>\n= 2520 \/ 2<br \/>\n= 1260<br \/>\nTherefore,<br \/>\nLCM(9, 10, 28) = 1260<\/p>\n<p>Question 41.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nAt Union Station, you notice that three subway lines just arrived at the same time. How long must you wait until all three lines arrive at Union Station at the same time again?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17050\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-80.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 80\" width=\"390\" height=\"251\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-80.png 390w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-80-300x193.png 300w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><\/p>\n<p>Answer: 60 minutes<\/p>\n<p>Explanation:<br \/>\nThe complete question in the attached figure<br \/>\nStep 1<br \/>\nFind the least common multiple (LCM) of the three numbers<br \/>\nList the prime factors of each number<br \/>\n10 = 2 \u00d7 5<br \/>\n12 = 2 \u00d7 2 \u00d7 3<br \/>\n15 = 3 \u00d7 5<br \/>\nMultiply each factor the greatest number of times it occurs in any of the numbers to find out the LCM<br \/>\nThe LCM is equal to<br \/>\n4 \u00d7 3 \u00d7 5 = 60<br \/>\nThus You must wait 60 minutes for all three lines to arrive at Union Station at the same time again.<\/p>\n<p>Question 42.<br \/>\n<strong>DIG DEEPER!<\/strong><br \/>\nA radio station gives away $15 to every 15th caller, $25 to every 25th caller, and a free concert ticket to every 100th caller. When will the station \ufb01rst give away all three prizes to one caller? When this happens, how much money and how many tickets are given away?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nRadio Station gives :<br \/>\n1st prize: $15 to 15th caller<br \/>\n2nd prize: $25 to 25th caller<br \/>\n3rd prize: free concert tickets to 100th caller<br \/>\nSo, in order to get all three prizes the caller must be 15th, 25th, and 100th caller at the same time. But to find when the radio station will give first all three prizes we calculate L.C.M. of ( 15, 25, 100 ) that is 300<br \/>\nHence, the station first gives away all three prizes to the 300th caller.<\/p>\n<p>Question 43.<br \/>\n<strong>LOGIC<\/strong><br \/>\nYou and a friend are running on treadmills. You run 0.5 mile every 3 minutes, and your friend runs 2 miles every 14 minutes. You both start and stop running at the same time and run a whole number of miles. What are the least possible numbers of miles you and your friend can run?<\/p>\n<p>Answer:<br \/>\nIf you run 0.5 miles every 3 minutes then you run 1 mile every 6 minutes.<br \/>\nIf your friend runs 2 miles every 14 minutes then your friend runs 1 mile every 7 minutes.<br \/>\nYou will both then both run a whole number of minutes for a time that is a multiple of 6 and 7.<br \/>\nThe least common multiple of 6 and 7 is 42 so the least possible time you and your friend could run for and both run a whole number of miles is then 42 minutes.<br \/>\nSince you run 1 mile every 6 minutes, in 42 minutes you will run 42\/6=7 miles.<br \/>\nSince your friend runs 1 mile every 7 minutes, in 42 minutes your friend will run 42\/7=6 miles.<\/p>\n<p>Question 44.<br \/>\n<strong>VENN DIAGRAM<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17051\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-81.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 81\" width=\"245\" height=\"156\" \/><br \/>\nRefer to the Venn diagram.<br \/>\na. Copy and complete the Venn diagram.<br \/>\nb. What is the LCM of 16, 24, and 40?<br \/>\nc. What is the LCM of 16 and 40? 24 and 40? 16 and 24? Explain how you found your answers.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-27763\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/01\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-81.jpg\" alt=\"Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-81\" width=\"245\" height=\"156\" \/><br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\n<strong>LCM of 16, 40:<\/strong><br \/>\nMultiples of 16:<br \/>\n16, 32, 48, 64, 80, 96, 112<br \/>\nMultiples of 40:<br \/>\n40, 80, 120, 160<br \/>\nTherefore,<br \/>\nLCM(16, 40) = 80<br \/>\n<strong>LCM of 24, 40:<\/strong><br \/>\nMultiples of 24:<br \/>\n24, 48, 72, 96, 120, 144, 168<br \/>\nMultiples of 40:<br \/>\n40, 80, 120, 160, 200<br \/>\nTherefore,<br \/>\nLCM(24, 40) = 120<br \/>\n<strong>LCM of 16, 24:<\/strong><br \/>\nMultiples of 16:<br \/>\n16, 32, 48, 64, 80<br \/>\nMultiples of 24:<br \/>\n24, 48, 72, 96<br \/>\nTherefore,<br \/>\nLCM(16, 24) = 48<\/p>\n<p><strong>CRITICAL THINKING<\/strong><br \/>\n<strong>Tell whether the statement is always, sometimes, or never true. Explain your reasoning.<\/strong><\/p>\n<p>Question 45.<br \/>\nThe LCM of two different prime numbers is their product.<\/p>\n<p>Answer: Always true<\/p>\n<p>Example:<br \/>\n3 \u00d7 5 = 15<\/p>\n<p>Question 46.<br \/>\nThe LCM of a set of numbers is equal to one of the numbers in the set.<\/p>\n<p>Answer:<br \/>\nThe LCM of a set of numbers is equal to one of the numbers in the set. Always Sometimes Never true. Question 909193: The LCM of a set of numbers is equal to one of the numbers in the set. This is sometimes true.<\/p>\n<p>Question 47.<br \/>\nThe GCF of two different numbers is the LCM of the numbers.<\/p>\n<p>Answer:<br \/>\nAnother way to find the LCM of two numbers is to divide their product by their greatest common factor ( GCF ). Example 2: Find the least common multiple of 18 and 20. The common factors are 2 and 3 .<\/p>\n<h3><a id=\"Numerical_Expressions_and_Factors_Connecting_Concepts\"><\/a>Numerical Expressions and Factors Connecting Concepts<\/h3>\n<p><strong>Getting Ready for Chapter Connecting Concepts<\/strong><\/p>\n<p><strong>Using the Problem-Solving Plan<\/strong><\/p>\n<p>Question 1.<br \/>\nA sports team gives away shirts at the stadium. There are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium. What is the greatest number of groups that can be formed using every shirt?<\/p>\n<p><strong>Understand the Problem<\/strong><br \/>\nYou know the number of large shirts and two relationships among the numbers of small, medium, and large shirts. You are asked to \ufb01nd the greatest number of identical groups that can be formed using every shirt.<br \/>\n<strong>Make a plan<\/strong><br \/>\nBreak the problem into parts. First use multiplication to \ufb01nd the number of each size shirt. Then \ufb01nd the GCF of these numbers.<br \/>\n<strong>Solve and Check<\/strong><br \/>\nUse the plan to solve the problem. Then check your solution.<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA sports team gives away shirts at the stadium.<br \/>\nThere are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium.<br \/>\n60 \u00d7 1.6 = 96<br \/>\n60 \u00d7 1.5 = 90<br \/>\nThe factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90<br \/>\nThe factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96<br \/>\nThen the greatest common factor is 6.<\/p>\n<p>Question 2.<br \/>\nAn escape artist \ufb01lls the tank shown with water. Find the number of cubic feet of water needed to \ufb01ll the tank. Then \ufb01nd the number of cubic yards of water that are needed to \ufb01ll the tank. Justify your answer.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16984\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-75.1.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 75.1\" width=\"154\" height=\"134\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nAn escape artist \ufb01lls the tank shown with water.<br \/>\nside = 6 ft<br \/>\nWe know that<br \/>\nThe volume of the cube = s\u00b3<br \/>\nV = 6ft \u00d7 6ft \u00d7 6ft<br \/>\nV = 216 cubic ft.<\/p>\n<p><strong>Performance Task<\/strong><\/p>\n<p><strong>Setting the Table<\/strong><\/p>\n<p>At the beginning of this chapter, you watched a STEAM video called \u201cFilling Pi\u00f1atas.\u201d You are now ready to complete the performance task for this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16989\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-82.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 82\" width=\"277\" height=\"203\" \/><\/p>\n<p>Answer:<br \/>\nFactors of 50 &#8211; 1, 2, 5, 10, 25, and 50<br \/>\nFactors of 12 &#8211; 1, 2, 3, 4, 6, 12<br \/>\nFactors of 16 &#8211; 1, 2, 4, 8, 16<br \/>\nThe factors of number 24 are 1, 2, 3, 4, 6, 8, 12, 24.<br \/>\nThe factors of 100 are 1,2,4,5,10,20,25,50 and 100.<\/p>\n<h3><a id=\"Numerical_Expressions_and_Factors_Chapter_Review\"><\/a>Numerical Expressions and Factors Chapter Review<\/h3>\n<p><strong>Review Vocabulary<\/strong><\/p>\n<p>Write the de\ufb01nition and give an example of each vocabulary term.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16990\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-83.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 83\" width=\"639\" height=\"111\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-83.png 639w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-83-300x52.png 300w\" sizes=\"(max-width: 639px) 100vw, 639px\" \/><\/p>\n<p><strong>Graphic Organizers<\/strong><\/p>\n<p>You can use an Information Frame to organize and remember concepts. Here is an example of an Information Frame for the vocabulary term power.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-16991\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-84.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 84\" width=\"541\" height=\"360\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-84.png 541w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-84-300x200.png 300w\" sizes=\"(max-width: 541px) 100vw, 541px\" \/><\/p>\n<p>Choose and complete a graphic organizer to help you study the concept.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16992\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-85.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 85\" width=\"308\" height=\"302\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-85.png 308w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-85-300x294.png 300w\" sizes=\"(max-width: 308px) 100vw, 308px\" \/><\/p>\n<ol>\n<li>perfect square<\/li>\n<li>numerical expression<\/li>\n<li>order of operations<\/li>\n<li>prime factorization<\/li>\n<li>greatest common factor (GCF)<\/li>\n<li>least common multiple (LCM)<\/li>\n<\/ol>\n<p>Answer:<br \/>\n1. A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system. Examples of Numbers that are Perfect Squares. 25 is a perfect square.<br \/>\n2. A numerical expression is a mathematical sentence involving only numbers and one or more operation symbols. Examples of operation symbols are the ones for addition, subtraction, multiplication, and division.<br \/>\n3. Order of operations refers to which operations should be performed in what order, but it&#8217;s just convention.<br \/>\n4. &#8220;Prime Factorization&#8221; is finding which prime numbers multiply together to make the original number.<br \/>\n5. Greatest Common Factor. The highest number that divides exactly into two or more numbers.<br \/>\n6. Least Common Multiple. The smallest positive number that is a multiple of two or more numbers.<\/p>\n<p><strong>Chapter Self-Assessment<\/strong><\/p>\n<p>As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16993\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-86.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 86\" width=\"635\" height=\"75\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-86.png 635w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-86-300x35.png 300w\" sizes=\"(max-width: 635px) 100vw, 635px\" \/><\/p>\n<p><strong>1.1 Powers and Exponents (pp. 3-8)<\/strong><\/p>\n<p>Write the product as a power.<\/p>\n<p>Question 1.<br \/>\n3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3<\/p>\n<p>Answer: The product of 3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 3 is 3<sup>6<\/sup><\/p>\n<p>Question 2.<br \/>\n5 \u00d7 5 \u00d7 5<\/p>\n<p>Answer: The product of 5 \u00d7 5 \u00d7 5 is 5<sup>3<\/sup><\/p>\n<p>Question 3.<br \/>\n17 . 17 . 17 . 17 . 17<\/p>\n<p>Answer: The product of 17 . 17 . 17 . 17 . 17 is 17<sup>5<\/sup><\/p>\n<p>Question 4.<br \/>\n3<sup>3<\/sup><\/p>\n<p>Answer: 3 \u00d7 3 \u00d7 3<\/p>\n<p>Question 5.<br \/>\n2<sup>6<\/sup><\/p>\n<p>Answer: 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2<\/p>\n<p>Question 6.<br \/>\n4<sup>4<\/sup><\/p>\n<p>Answer: 4 \u00d7 4 \u00d7 4 \u00d7 4<\/p>\n<p>Question 7.<br \/>\nWrite a power that has a value greater than 2<sup>3<\/sup> and less than 3<sup>3<\/sup>.<\/p>\n<p>Answer: The power that has a value greater than 2<sup>3<\/sup> and less than 3<sup>3<\/sup> is 4\u00b2<\/p>\n<p>Question 8.<br \/>\nWithout evaluating, determine whether 2<sup>5<\/sup> or 4<sup>2<\/sup> is greater. Explain.<\/p>\n<p>Answer: 2<sup>5<\/sup> &gt; 4<sup>2<\/sup><\/p>\n<p>Explanation:<br \/>\nThe exponent with the highest number will be greater.<\/p>\n<p>Question 9.<br \/>\nThe bases on a softball \ufb01eld are square. What is the area of each base?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17052\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-87.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 87\" width=\"135\" height=\"111\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17053\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-88.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 88\" width=\"420\" height=\"279\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-88.png 420w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-88-300x199.png 300w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nThe bases on a softball \ufb01eld are square.<br \/>\ns = 15 inches<br \/>\nWe know that,<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 15 \u00d7 15<br \/>\nA = 225 sq. in<br \/>\nThus the area of each base is 225 sq. in.<\/p>\n<p><strong>1.2 Order of Operations (pp. 9\u201314)<\/strong><\/p>\n<p><strong>Evaluate the expression.<\/strong><\/p>\n<p>Question 10.<br \/>\n3 \u00d7 6 &#8211; 12 \u00f7 6<\/p>\n<p>Answer: 16<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n(3 \u00d7 6) &#8211; (12 \u00f7 6)<br \/>\n18 &#8211; 2 = 16<\/p>\n<p>Question 11.<br \/>\n30 \u00f7 (14 &#8211; 2<sup>2<\/sup>) \u00d7 5<\/p>\n<p>Answer: 15<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n30 \u00f7 (14 &#8211; 4) \u00d7 5<br \/>\n30 \u00f7 10 \u00d7 5<br \/>\n3 \u00d7 5 = 15<\/p>\n<p>Question 12.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17054\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-89.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 89\" width=\"93\" height=\"46\" \/><\/p>\n<p>Answer: 15<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n2.3 + 3.7 = 6<br \/>\n5(6)\/2 = 30\/2 = 15<\/p>\n<p>Question 13.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17027\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-6th-Grade-Chapter-1-Numerical-Numerical-Expressions-and-Factors-90.png\" alt=\"Big Ideas Math Answers 6th Grade Chapter 1 Numerical Expressions and Factors 90\" width=\"108\" height=\"42\" \/><\/p>\n<p>Answer: 37<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n7\u00b2 + 5 = 49 + 5 = 54<br \/>\n1\/2 \u00d7 54 = 27<br \/>\n4\u00b3 &#8211; 27 = 64 &#8211; 27 = 37<\/p>\n<p>Question 14.<br \/>\n20 (3<sup>2<\/sup> &#8211; 4) \u00f7 50<\/p>\n<p>Answer: 2<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n(3<sup>2<\/sup> &#8211; 4) = 9 &#8211; 4 = 5<br \/>\n20 \u00d7 5 \u00f7 50<br \/>\n100 \u00f7 50 = 2<\/p>\n<p>Question 15.<br \/>\n5 + 3(4<sup>2<\/sup> &#8211; 2) \u00f7 6<\/p>\n<p>Answer: 12<\/p>\n<p>Explanation:<br \/>\nYou have to evaluate from left to right.<br \/>\n(4<sup>2<\/sup> &#8211; 2) = 16 &#8211; 2 = 14<br \/>\n5 + 3(14) \u00f7 6<br \/>\n5 + 42 \u00f7 6<br \/>\n5 + 7 = 12<\/p>\n<p>Question 16.<br \/>\nUse grouping symbols and at least one exponent to write a numerical expression that has a value of 80.<\/p>\n<p>Answer: 6 + (9\u00b2 &#8211; 7) = 80<\/p>\n<p><strong>1.3 Prime Factorization (pp. 15\u201320)<\/strong><\/p>\n<p>List the factor pairs of the number.<\/p>\n<p>Question 17.<br \/>\n28<\/p>\n<p>Answer: The factor pairs of the number 28 are 1, 2, 4, 7, 14, 28<\/p>\n<p>Explanation:<br \/>\n28 = 1 \u00d7 28<br \/>\n2 \u00d7 14<br \/>\n4 \u00d7 7<br \/>\n7 \u00d7 4<br \/>\n14 \u00d7 2<br \/>\n28 \u00d7 1<\/p>\n<p>Question 18.<br \/>\n44<\/p>\n<p>Answer: The factor pairs of the number 44 are 1, 2, 4, 11, 44.<\/p>\n<p>Explanation:<br \/>\n44 = 1 \u00d7 44<br \/>\n2 \u00d7 22<br \/>\n4 \u00d7 11<br \/>\n11 \u00d7 4<br \/>\n44 \u00d7 1<\/p>\n<p>Question 19.<br \/>\n96<\/p>\n<p>Answer: The factor pairs of the number 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96<\/p>\n<p>Explanation:<br \/>\n1 and 96 are a factor pair of 96 since 1 x 96= 96<br \/>\n2 and 48 are a factor pair of 96 since 2 x 48= 96<br \/>\n3 and 32 are a factor pair of 96 since 3 x 32= 96<br \/>\n4 and 24 are a factor pair of 96 since 4 x 24= 96<br \/>\n6 and 16 are a factor pair of 96 since 6 x 16= 96<br \/>\n8 and 12 are a factor pair of 96 since 8 x 12= 96<br \/>\n12 and 8 are a factor pair of 96 since 12 x 8= 96<br \/>\n16 and 6 are a factor pair of 96 since 16 x 6= 96<br \/>\n24 and 4 are a factor pair of 96 since 24 x 4= 96<br \/>\n32 and 3 are a factor pair of 96 since 32 x 3= 96<br \/>\n48 and 2 are a factor pair of 96 since 48 x 2= 96<br \/>\n96 and 1 are a factor pair of 96 since 96 x 1= 96<\/p>\n<p>Question 20.<br \/>\nThere are 36 graduated cylinders to put away on a shelf after science class. The shelf can \ufb01t a maximum of 20 cylinders across and 4 cylinders deep. The teacher wants each row to have the same number of cylinders. List the possible arrangements of the graduated cylinders on the shelf.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17055\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-91.png\" alt=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors 91\" width=\"256\" height=\"176\" \/><\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nThere are 36 graduated cylinders to put away on a shelf after science class.<br \/>\nThe shelf can \ufb01t a maximum of 20 cylinders across and 4 cylinders deep.<br \/>\nThe teacher wants each row to have the same number of cylinders.<br \/>\nThere are three possible arrangements of the graduated cylinders on the shelf.<br \/>\n1. 4 rows of 9 graduated cylinders<br \/>\n2. 3 rows of 12 graduated cylinders.<br \/>\n3. 2 rows of 18 graduated cylinders.<\/p>\n<p><strong>Write the prime factorization of the number.<\/strong><\/p>\n<p>Question 21.<br \/>\n42<\/p>\n<p>Answer: The prime factorization of the number 42 is 2 \u00d7 3 \u00d7 7<\/p>\n<p>Explanation:<br \/>\n42 = 2 \u00d7 21<br \/>\n= 2 \u00d7 3 \u00d7 7<\/p>\n<p>Question 22.<br \/>\n50<\/p>\n<p>Answer: The prime factorization of the number\u00a0 2 \u00d7 5 \u00d7 5<\/p>\n<p>Explanation:<br \/>\n50 = 2 \u00d7 25<br \/>\n= 2 \u00d7 5 \u00d7 5<\/p>\n<p>Question 23.<br \/>\n66<\/p>\n<p>Answer: The prime factorization of the number 2 \u00d7 3 \u00d7 11<\/p>\n<p>Explanation:<br \/>\n66 = 2 \u00d7 33<br \/>\n= 2 \u00d7 3 \u00d7 11<\/p>\n<p><strong>1.4 Greatest Common Factor (pp. 21\u201326)<\/strong><\/p>\n<p><strong>Find the GCF of the numbers using lists of factors.<\/strong><\/p>\n<p>Question 24.<br \/>\n27, 45<\/p>\n<p>Answer: GCF is 9<\/p>\n<p>Explanation:<br \/>\nThe factors of 27 are: 1, 3, 9, 27<br \/>\nThe factors of 45 are: 1, 3, 5, 9, 15, 45<br \/>\nThen the greatest common factor is 9.<\/p>\n<p>Question 25.<br \/>\n30, 48<\/p>\n<p>Answer: GCF is 6<\/p>\n<p>Explanation:<br \/>\nThe factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30<br \/>\nThe factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48<br \/>\nThen the greatest common factor is 6.<\/p>\n<p>Question 26.<br \/>\n28, 48<\/p>\n<p>Answer: GCF is 4<\/p>\n<p>Explanation:<br \/>\nThe factors of 28 are: 1, 2, 4, 7, 14, 28<br \/>\nThe factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48<br \/>\nThen the greatest common factor is 4.<\/p>\n<p><strong>Find the GCF of the numbers using prime factorizations.<\/strong><\/p>\n<p>Question 27.<br \/>\n24, 80<\/p>\n<p>Answer:<br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 24 is 24 = 2 \u00b7 2 \u00b7 2 \u00b7 3 = 2\u00b3 . 3<br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 80 is 80 = 2 x 2 x 2 x 2 x 5 = 2\u00b2 . 2\u00b2 . 5<\/p>\n<p>Question 28.<br \/>\n52, 68<\/p>\n<p>Answer:<br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 52 is 2 x 2 x 13 = 2\u00b2 . 13<br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 68 is 2 \u00d7 2 \u00d7 17 = 2\u00b2. 17<\/p>\n<p>Question 29.<br \/>\n32, 56<\/p>\n<p>Answer:<br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 32 is 2 x 2 x 2 x 2 x 2 = 2<sup>5<\/sup><br \/>\nThe prime factorization is the product of the circled primes. So the prime factorization of 56 is 2 x 2 x 2 x 7 = 2\u00b3 . 7<\/p>\n<p>Question 30.<br \/>\nWrite a pair of numbers that have a GCF of 20.<\/p>\n<p>Answer: The prime factors of 20 are 2 x 2 x 5. The GCF of 20 is 5.<\/p>\n<p>Question 31.<br \/>\nWhat is the greatest number of friends you can invite to an arcade using the coupon such that the tokens and slices of pizza are equally split between you and your friends with none left over? How many slices of pizza and tokens will each person receive?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16994\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-92.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 92\" width=\"284\" height=\"211\" \/><\/p>\n<p>Answer: (n\/4)-1<\/p>\n<p>Explanation:<br \/>\nTotal slices = n<br \/>\nTotal number of people =4<br \/>\nEach people may be eat = n\/4 slices<br \/>\nHere Harris eats 1 slice fewer<br \/>\nThen Harris eats (n\/4)-1 slices<\/p>\n<p><strong>1.5 Least Common Multiple (pp. 27\u201332)<\/strong><\/p>\n<p>Find the LCM of the numbers using lists of multiples.<\/p>\n<p>Question 32.<br \/>\n4, 14<\/p>\n<p>Answer: 28<\/p>\n<p>Explanation:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 4:<br \/>\n4, 8, 12, 16, 20, 24, 28, 32, 36<br \/>\nMultiples of 14:<br \/>\n14, 28, 42, 56<br \/>\nTherefore,<br \/>\nLCM(4, 14) = 28<\/p>\n<p>Question 33.<br \/>\n6, 20<\/p>\n<p>Answer: 60<\/p>\n<p>Explanation:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 6:<br \/>\n6, 12, 18, 24, 30, 36, 42, 48, 54, 60.<br \/>\nMultiples of 20:<br \/>\n20, 40, 60, 80, 100<br \/>\nThe LCM of 6, 20 is 60<\/p>\n<p>Question 34.<br \/>\n12, 28<\/p>\n<p>Answer: 84<\/p>\n<p>Explanation:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60, 72, 84, 96<br \/>\nMultiples of 28:<br \/>\n28, 56, 84, 112, 140, 168, 196<br \/>\nLCM of 12, 28 is 84<\/p>\n<p><strong>Find the LCM of the numbers using prime factorizations.<\/strong><\/p>\n<p>Question 35.<br \/>\n6, 45<\/p>\n<p>Answer:<br \/>\nPrime Factorisation of 6: 2 \u00d7 3<br \/>\nPrime Factorisation of 45: 3 \u00d7 3 \u00d7 5<br \/>\nLCM is 2 \u00d7 3 \u00d7 3 \u00d7 3 \u00d7 5 = 60<\/p>\n<p>Question 36.<br \/>\n10, 12<\/p>\n<p>Answer: 60<br \/>\nPrime factorization of 10: 2 \u00d7 5<br \/>\nPrime factorization of 12: 2 \u00d7 2 \u00d7 3<br \/>\nLCM is 5 \u00d7 2 \u00d7 2 \u00d7 3 = 60<\/p>\n<p>Question 37.<br \/>\n18, 27<\/p>\n<p>Answer:<br \/>\nPrime factorization of 18: 2 \u00d7 3 \u00d7 3<br \/>\nPrime factorization of 27: 3 \u00d7 3 \u00d7 3<br \/>\nLCM is 2 \u00d7 3 \u00d7 3 \u00d7 3 = 54<\/p>\n<p>Question 38.<br \/>\nFind the LCM of 8, 12, and 18.<\/p>\n<p>Answer: 72<br \/>\nPrime Factorisation of 8: 2 \u00d7 2 \u00d7 2<br \/>\nPrime Factorisation of 12: 2 \u00d7 2 \u00d7 3<br \/>\nPrime factorization of 18: 2 \u00d7 3 \u00d7 3<br \/>\nLCM = 72<\/p>\n<p>Question 39.<br \/>\nWrite a pair of numbers that have an LCM of 84.<\/p>\n<p>Answer: 84 and 12<\/p>\n<p>Explanation:<br \/>\nThe LCM of 84 and 12 is 84.<br \/>\nPrime factorization of 12: 2 \u00d7 2 \u00d7 3<br \/>\nPrime factorization of 84: 2 \u00d7 2 \u00d7 3 \u00d7 7<br \/>\nThe Least Common Multiple is 2 \u00d7 2 \u00d7 3 \u00d7 7 = 84<\/p>\n<p>Question 40.<br \/>\nWrite three numbers that have an LCM of 45.<\/p>\n<p>Answer: 3, 15, 45<\/p>\n<p>Explanation:<br \/>\nThe prime factorization of 15: 3 \u00d7 5<br \/>\nThe prime factorization of 45: 3 \u00d7 3 \u00d7 5<br \/>\nThe LCM of 3, 15, 45 is 45.<\/p>\n<p>Question 41.<br \/>\nYou water your roses every sixth day and your hydrangeas every \ufb01fth day. Today you water both plants. In how many days will you water both plants on the same day again?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16995\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-93.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 93\" width=\"348\" height=\"273\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-93.png 348w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-93-300x235.png 300w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p>\n<p>Answer: 30<\/p>\n<p>Explanation:<br \/>\nGiven,<br \/>\nYou water your roses every sixth day and your hydrangeas every \ufb01fth day. Today you water both plants.<br \/>\n6 \u00d7 5 = 30<br \/>\nYou water both plants for 30 days on the same day again.<\/p>\n<p>Question 42.<br \/>\nHamburgers are sold in packages of 20, while buns are sold in packages of 12. What are the least numbers of packages you should buy in order to have the same number of hamburgers and buns?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nHamburgers are sold in packages of 20, while buns are sold in packages of 12.<br \/>\nAt least 5 packages of buns and 3 packages of hamburgers.<br \/>\n20\u00d73=60<br \/>\n12\u00d75=60<br \/>\nSo that is how you get the answer by seeing if they have any integers in common.<\/p>\n<p>Question 43.<br \/>\nA science museum is giving away a magnetic liquid kit to every 50th guest and a plasma ball to every 35th guest until someone receives both prizes.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16996\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-94.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 94\" width=\"209\" height=\"303\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-94.png 209w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-94-207x300.png 207w\" sizes=\"(max-width: 209px) 100vw, 209px\" \/><br \/>\na. Which numbered guest will receive both a magnetic liquid kit and a plasma ball?<\/p>\n<p>Answer:<br \/>\nA magnetic liquid kit prize every 50 guests and a plasma ball every 35 guests.<br \/>\n1.Guest 50th<br \/>\n2.Guest 100th<br \/>\n3.Guest 150th<br \/>\n4.Guest 200th<br \/>\n5.Guest 250th<br \/>\n6.Guest 300th<br \/>\n7.Guest 350th<br \/>\n8.Guest 400th and so on, in case no coincidence would happen.<\/p>\n<p>b. How many people will receive a plasma ball?<\/p>\n<p>Answer:<br \/>\n1.Guest 35th<br \/>\n2.Guest 70th<br \/>\n3.Guest 105th<br \/>\n4.Guest 140th<br \/>\n5.Guest 175th<br \/>\n6.Guest 210th<br \/>\n7.Guest 245th<br \/>\n8.Guest 280th<br \/>\n9.Guest 315th<br \/>\n10.Guest 350th.<br \/>\nAs you can see, Guest 350th will be the first one to receive both prizes, and including him or her, a total of ten guests will receive the plasma ball until that moment. There wasn&#8217;t any coincidence before Guest 350th.<\/p>\n<h3><a id=\"Numerical_Expressions_and_Factors_Practice_Test\"><\/a>Numerical Expressions and Factors Practice Test<\/h3>\n<p>Question 1.<br \/>\nFind the value of 2<sup>3<\/sup>.<\/p>\n<p>Answer:<br \/>\n2<sup>3<\/sup> can be written as 2 \u00d7 2 \u00d7 2 = 8<br \/>\nThus the value of 2<sup>3 <\/sup>is 8.<\/p>\n<p>Question 2.<br \/>\nEvaluate <img loading=\"lazy\" class=\"alignnone size-full wp-image-16997\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-95.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 95\" width=\"95\" height=\"44\" \/><\/p>\n<p>Answer:<br \/>\n5 + 4(12 &#8211; 2) = 5 + 4(10) = 5 + 40<br \/>\n(5 + 40)\/3\u00b2 = 45\/9 = 5<br \/>\nThus the value of <img loading=\"lazy\" class=\"alignnone size-full wp-image-16997\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-95.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 95\" width=\"95\" height=\"44\" \/>is 5.<\/p>\n<p>Question 3.<br \/>\nWrite 264.264.264 as a power<\/p>\n<p>Answer: 264.264.264 can be written as 264\u00b3<\/p>\n<p>Question 4.<br \/>\nList the factor pairs of 66.<\/p>\n<p>Answer: The factor pairs of 66 are (1,66) (2, 33) (6, 11)<br \/>\n66 = 1 \u00d7 66<br \/>\n66 = 2 \u00d7 33<br \/>\n66 = 6 \u00d7 11<\/p>\n<p>Question 5.<br \/>\nWrite the prime factorization of 56.<\/p>\n<p>Answer:<br \/>\n56 = 2 \u00d7 28<br \/>\n= 2 \u00d7 2 \u00d7 14<br \/>\n= 2 \u00d7 2 \u00d7 2 \u00d7 7<br \/>\nThus the prime factorization of 56 is\u00a02 \u00d7 2 \u00d7 2 \u00d7 7<\/p>\n<p>Find the GCF of the numbers.<\/p>\n<p>Question 6.<br \/>\n24, 54<\/p>\n<p>Answer: GCF is 6<\/p>\n<p>The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24<br \/>\nThe factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54<br \/>\nThen the greatest common factor is 6.<\/p>\n<p>Question 7.<br \/>\n16, 32, 72<\/p>\n<p>Answer: GCF is 8<\/p>\n<p>The factors of 16 are: 1, 2, 4, 8, 16<br \/>\nThe factors of 32 are: 1, 2, 4, 8, 16, 32<br \/>\nThe factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72<br \/>\nThen the greatest common factor is 8.<\/p>\n<p>Question 8.<br \/>\n52, 65<\/p>\n<p>Answer: GCF is 13<\/p>\n<p>The factors of 52 are: 1, 2, 4, 13, 26, 52<br \/>\nThe factors of 65 are: 1, 5, 13, 65<br \/>\nThen the greatest common factor is 13.<\/p>\n<p><strong>Find the LCM of the numbers.<\/strong><\/p>\n<p>Question 9.<br \/>\n9, 24<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 9:<br \/>\n9, 18, 27, 36, 45, 54, 63, 72, 81, 90<br \/>\nMultiples of 24:<br \/>\n24, 48, 72, 96, 120<br \/>\nTherefore,<br \/>\nLCM(9, 24) = 72<\/p>\n<p>Question 10.<br \/>\n26, 39<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 26:<br \/>\n26, 52, 78, 104, 130<br \/>\nMultiples of 39:<br \/>\n39, 78, 117, 156<br \/>\nTherefore,<br \/>\nLCM(26, 39) = 78<\/p>\n<p>Question 11.<br \/>\n6, 12, 14<\/p>\n<p>Answer:<br \/>\nFind and list multiples of each number until the first common multiple is found. This is the lowest common multiple.<br \/>\nMultiples of 6:<br \/>\n6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96<br \/>\nMultiples of 12:<br \/>\n12, 24, 36, 48, 60, 72, 84, 96, 108<br \/>\nMultiples of 14:<br \/>\n14, 28, 42, 56, 70, 84, 98, 112<br \/>\nTherefore,<br \/>\nLCM(6, 12, 14) = 84<\/p>\n<p>Question 12.<br \/>\nYou have 16 yellow beads, 20 red beads, and 24 orange beads to make identical bracelets. What is the greatest number of bracelets that you can make using all of the beads?<\/p>\n<p>Answer:<br \/>\nTo find how many identical bracelets you can make, you need to find a common denominator.<br \/>\nIn this case, all three numbers; 16, 20, and 24, can be divided by four.<br \/>\nSo you now know you can have four bracelets.<br \/>\nThen you take your numbers of each color beads and divide them by four so you know how many of each color will be on the bracelets.<br \/>\nIn the end, you have four bracelets, each with 4 yellow beads, 5 red beads and 6 orange beads<\/p>\n<p>Question 13.<br \/>\nA bag contains equal numbers of green marbles and blue marbles. You can divide all of the green marbles into groups of 12 and all the blue marbles into groups of 16. What is the least number of each color of marble that can be in the bag?<\/p>\n<p>Answer:<br \/>\nGiven,<br \/>\nA bag contains equal numbers of green marbles and blue marbles.<br \/>\nYou can divide all of the green marbles into groups of 12 and all the blue marbles into groups of 16.<br \/>\nTo solve this problem, we need to find for the LCM of each number. That is:<br \/>\n12: 12, 24, 36, 48, 60<br \/>\n16: 16, 32, 48, 64, 80<br \/>\nSo we can see that the LCM is 48.<br \/>\nTherefore the least number of each color of marble must be 48.<\/p>\n<p>Question 14.<br \/>\nThe ages of the members of a family are 65, 58, 27, 25, 5, and 2 years old. What is the total admission price for the family to visit the zoo?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17000\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-96.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 96\" width=\"231\" height=\"161\" \/><\/p>\n<p>Answer:<br \/>\nThe ages of the members of a family are 65, 58, 27, 25, 5, and 2 years old.<br \/>\nWe can find the total admission price for the family to visit the zoo by following the above table.<br \/>\n$10 + $12 + $27 + $12 + $8 + $8 = $77<\/p>\n<p>Question 15.<br \/>\nA competition awards prizes for fourth, third, second, and \ufb01rst place. The fourth place winner receives $5. Each place above that receives a prize that is \ufb01ve times the amount of the previous prize. How much prize money is awarded?<\/p>\n<p>Answer:<br \/>\nA competition awards prizes for fourth, third, second, and \ufb01rst place. The fourth place winner receives $5. Each place above that receives a prize that is \ufb01ve times the amount of the previous prize.<br \/>\nEach place above that receives a prize that is five times the amount of the previous prize<br \/>\nSo we can say that;<br \/>\nPn = 5 \u00d7\u00a0 P(n+1)<br \/>\nWhere n \u21d2 Number Place<br \/>\nPn = Price received by Number place<br \/>\nSubstituting the values of n as 3,2,1 to find the price of third second first place winner.<br \/>\nP3 = 5 \u00d7\u00a0 P(3+1) = 5 \u00d7 P4 = 5 \u00d7 5 = 25<br \/>\nP2 = 5 \u00d7\u00a0 P(2+1) = 5 \u00d7 P3 = 5 \u00d7 25 = 125<br \/>\nP1 = 5 \u00d7\u00a0 P(1+1) = 5 \u00d7 P2 = 5 \u00d7 125 = 625<br \/>\nNow We will find the Total Prize money awarded.<br \/>\nTotal Prize money awarded = 625 + 125 + 25 + 5 = 780<br \/>\nHence A total of $780 price money was awarded.<\/p>\n<p>Question 16.<br \/>\nYou buy tealight candles and mints as party favors for a baby shower. The tealight candles come in packs of 12 for $3.50. The mints come in packs of 50 for $6.25. What is the least amount of money you can spend to buy the same number of candles and mints?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16998\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-96.1.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 96.1\" width=\"304\" height=\"172\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-96.1.png 304w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-96.1-300x170.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/p>\n<p>Answer: The least amount of money it can be spent is $125.<\/p>\n<p>Explanation:<br \/>\nFirst, we write the prime factorization of each number:<br \/>\n12= 2\u00b72\u00b73<br \/>\n15= 2\u00b75\u00b75<br \/>\nThen, we search for each different factor which appears the greater number of times. The factor 2 appears in both factorizations so the least common multiple is:<br \/>\nLCM= 2\u00b72\u00b73\u00b75\u00b75=300<br \/>\nHence, the total quantity of packs of each thing is:<br \/>\nCandles: 300\u00f712=25<br \/>\nMints: 300\u00f750=6<br \/>\nThe least amount of money it can be spent is:<br \/>\nT=25\u00d7$3.50 + 6\u00d7$6.25= $87.5 + $37.5= $125<\/p>\n<h3><a id=\"Numerical_Expressions_and_Factors_Cumulative_Practice\"><\/a>Numerical Expressions and Factors Cumulative Practice<\/h3>\n<p>Question 1.<br \/>\nFind the value of 8 \u00d7 135?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-16999\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-96.2.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 96.2\" width=\"86\" height=\"70\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17001\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-97.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 97\" width=\"334\" height=\"391\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-97.png 334w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-97-256x300.png 256w\" sizes=\"(max-width: 334px) 100vw, 334px\" \/><\/p>\n<p>Answer: Multiply the two numbers 8 and 135<br \/>\n8 \u00d7 135 = 1080<\/p>\n<p>Question 2.<br \/>\nWhich number is equivalent to the expression blow?<br \/>\n3.2<sup>3<\/sup> &#8211; 8 \u00f7 4<\/p>\n<p>Answer: 22<\/p>\n<p>Explanation:<br \/>\nGiven the expression 3.2<sup>3<\/sup> &#8211; 8 \u00f7 4<br \/>\n3 \u00d7 8 -(8\u00f74)<br \/>\n24 &#8211; (2)<br \/>\n24 &#8211; 2 = 22<\/p>\n<p>Question 3.<br \/>\nThe top of an end table is a square with a side length of 16 inches. What is the area of the tabletop?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17002\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-98.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 98\" width=\"369\" height=\"315\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-98.png 369w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-98-300x256.png 300w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><\/p>\n<p>Answer: I<\/p>\n<p>Explanation:<br \/>\nGiven that<br \/>\nThe top of an end table is a square with a side length of 16 inches.<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 16 \u00d7 16<br \/>\nA = 256 in\u00b2<br \/>\nThus the correct answer is option I.<\/p>\n<p>Question 4.<br \/>\nYou are \ufb01lling baskets using 18 green eggs, 36 red eggs, and 54 blue eggs. What is the greatest number of baskets that you can \ufb01ll so that the baskets are identical and there are no eggs left over?<br \/>\nA. 3<br \/>\nB. 6<br \/>\nC. 9<br \/>\nD. 18<\/p>\n<p>Answer: D<\/p>\n<p>Explanation:<br \/>\nGiven,<br \/>\nYou are \ufb01lling baskets using 18 green eggs, 36 red eggs, and 54 blue eggs.<br \/>\n18\/n = 36\/n = 54\/n<br \/>\nFactors of 18 are 2,3,6,9,18<br \/>\nFactors of 36 are 2,3,4,6,9,12,18.<br \/>\nFactors of 54 are 2,3,6,9,18,27,54.<br \/>\nThe common multiples of 18,36 and 54 are 2,3,6,9,18.<br \/>\nThus the greatest among them is 18.<br \/>\nThus the correct answer is option D.<\/p>\n<p>Question 5.<br \/>\nWhat is the value of 2<sup>3<\/sup>.3<sup>2<\/sup>.5?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17003\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-99.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 99\" width=\"89\" height=\"71\" \/><\/p>\n<p>Answer:<br \/>\n2<sup>3<\/sup>.3<sup>2<\/sup>.5<br \/>\n2\u00b3 = 8<br \/>\n3\u00b2 = 9<br \/>\n8 \u00d7 9 \u00d7 5 = 360<\/p>\n<p>Question 6.<br \/>\nYou hang the two strands of decorative lights shown below.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17004\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-100.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 100\" width=\"527\" height=\"152\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-100.png 527w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-100-300x87.png 300w\" sizes=\"(max-width: 527px) 100vw, 527px\" \/><br \/>\nBoth strands just changed color. After how many seconds will the strands change color at the same time again?<br \/>\nF. 3 seconds<br \/>\nG. 30 seconds<br \/>\nH. 90 seconds<br \/>\nI. 270 seconds<\/p>\n<p>Answer: 3 seconds<\/p>\n<p>Explanation:<\/p>\n<p>Strand I: Changes between red and blue every 15 seconds<br \/>\nStrand II: Changes between green and gold every 18 seconds<br \/>\n18 &#8211; 15 = 3 seconds<br \/>\nThus the correct answer is option F.<\/p>\n<p>Question 7.<br \/>\nPoint P is plotted in the coordinate plane below.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17005\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-101.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 101\" width=\"181\" height=\"176\" \/><br \/>\nWhat are the coordinates of Point P ?<br \/>\nA. (5, 3)<br \/>\nB. (4, 3)<br \/>\nC. (3, 5)<br \/>\nD. (3, 4)<\/p>\n<p>Answer: C<br \/>\nBy seeing the above graph we can find the coordinates of point p.<br \/>\nThe X-axis is on 3 and Y-axis is on 5.<br \/>\nThus the correct answer is option c.<\/p>\n<p>Question 8.<br \/>\nWhat is the prime factorization of 1100?<br \/>\nF. 2 \u00d7 5 \u00d7 11<br \/>\nG. 2<sup>2<\/sup> \u00d7 5<sup>2<\/sup> \u00d7 11<br \/>\nH. 4 \u00d7 5<sup>2<\/sup> \u00d7 11<br \/>\nI. 2<sup>2<\/sup> \u00d7 5 \u00d7 55<\/p>\n<p>Answer: G<br \/>\nPrime factorization of 1100: 2 \u00d7 550<br \/>\n2 \u00d7 2 \u00d7 275<br \/>\n2 \u00d7 2 \u00d7 5 \u00d7 55<br \/>\n2 \u00d7 2 \u00d7 5 \u00d7 5 \u00d7 11<br \/>\nThus the correct answer is option b.<\/p>\n<p>Question 9.<br \/>\nWhat is the least common multiple of 3, 8, and 10?<br \/>\nA. 24<br \/>\nB. 30<br \/>\nC. 80<br \/>\nD. 120<\/p>\n<p>Answer: D<\/p>\n<p>Multiples of 3:<br \/>\n3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126<br \/>\nMultiples of 8:<br \/>\n8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136<br \/>\nMultiples of 10:<br \/>\n10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 140.<br \/>\nThe common multiple among the three is 120.<br \/>\nThus the correct answer is option D.<\/p>\n<p>Question 10.<br \/>\nWhat is the area of the shaded region of the \ufb01gure below?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17006\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-102.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 102\" width=\"357\" height=\"283\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-102.png 357w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-102-300x238.png 300w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><\/p>\n<p>Answer:<br \/>\nThe above figure is square.<br \/>\ns = 4 yd<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 4 yd \u00d7 4 yd<br \/>\nA = 16 sq. yd<br \/>\nThe area of the outer box.<br \/>\ns = 9 yd<br \/>\nArea of the square = s \u00d7 s<br \/>\nA = 9 yd \u00d7 9 yd<br \/>\nA = 81 sq. yd<br \/>\nThe area of the shaded region is 81 &#8211; 16 = 65 sq. yd<br \/>\nThus the correct answer is option G.<\/p>\n<p>Question 11.<br \/>\nWhich expression represents a prime factorization?<br \/>\nA. 4 \u00d7 4 \u00d7 7<br \/>\nB. 2<sup>2<\/sup> \u00d7 21 \u00d7 23<br \/>\nC. 3<sup>4<\/sup> \u00d7 5 \u00d7 7<br \/>\nD. 5 \u00d7 5 \u00d7 9 \u00d7 11<\/p>\n<p>Answer: B<\/p>\n<p>Prime factorization:<br \/>\n2<sup>2<\/sup> \u00d7 21 \u00d7 23<br \/>\n2, 21, 23 is a prime number.<br \/>\nThus the correct answer is option B.<\/p>\n<p>Question 12.<br \/>\nFind the greatest common factor for each pair of numbers.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-17007\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2020\/12\/Big-Ideas-Math-Answer-Key-Grade-6-Chapter-1-Numerical-Numerical-Expressions-and-Factors-103.png\" alt=\"Big Ideas Math Answer Key Grade 6 Chapter 1 Numerical Expressions and Factors 103\" width=\"73\" height=\"65\" \/><br \/>\nWhat can you conclude about the greatest common factor of 10, 15, and 21? Explain your reasoning.<\/p>\n<p>Answer: 1<\/p>\n<p>Explanation:<br \/>\nThe factors of 10 are: 1, 2, 5, 10<br \/>\nThe factors of 15 are: 1, 3, 5, 15<br \/>\nThe factors of 21 are: 1, 3, 7, 21<br \/>\nThen the greatest common factor is 1.<\/p>\n<p>Final Words:<br \/>\nI wish that the details prevailed in this article regarding Big Ideas Math Grade 6 Chapter 1 Numerical Expressions and Factors Answer Key is helpful for all the students and also teachers. Make use of the solutions and score good marks in the exams. Best Of Luck!!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors: Before knowing the numerical expressions, Grade 6 students must be familiar with composite numbers, factor pairs, and prime numbers. Refer to the concept of vocabulary terms, subtracting and adding the fractions, mixed fractions in the below sections. Know the various terms like what &#8230; <a title=\"Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors\" class=\"read-more\" href=\"https:\/\/bigideasmathanswers.com\/big-ideas-math-answers-grade-6-chapter-1\/\" aria-label=\"More on Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":89738,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[4],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/04\/Big-Ideas-Math-Answers-Grade-6-Chapter-1-Numerical-Expressions-and-Factors.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/16765"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=16765"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/16765\/revisions"}],"predecessor-version":[{"id":90239,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/16765\/revisions\/90239"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/89738"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=16765"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=16765"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=16765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}