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1.3Prime FactorizationLearning Target:Write a number as a product of prime factors and represent the product using exponents.Success Criteria: EXPLORATIONEXPLOORATION 1I can find factor pairs of a number.I can explain the meanings of prime and composite numbers.I can create a factor tree to find the prime factors of a number.I can write the prime factorization of a number.Rewriting Numbers as Products of FactorsWork with a partner. Two students use factor trees to write 108 as aproduct of factors, as shown below.Student AStudent Ba. Without using 1 as a factor, can you write 108 as a product withmore factors than each student used? Justify your answer.Math PracticeInterpret ResultsHow do you know youranswer makes sense?b. Use factor trees towrite 80, 162, and300 as products of asmany factors as possible.Do not use 1 as a factor.808c. Compare your resultsin parts (a) and (b) withother groups. For eachnumber, identify the productwith the greatest number of factors.What do these factors have in common?Section 1.3ms2019 gr6 Ch01.indb 15Prime Factorization151/16/18 4:27 PM

1.3Lesson Because 2 is a factor of 10 and 2 5 10, 5 is also a factor of 10. The pair 2, 5 iscalled a factor pair of 10.EXAMPLE1 Finding Factor PairsThe brass section of a marching band has30 members. The band director arranges thebrass section in rows. Each row has the samenumber of members. How many possiblearrangements are there?Key Vocabularyfactor pair, p. 16prime factorization,p. 16factor tree, p. 16Use the factor pairs of 30 to find the number of arrangements. 30 2 1530 3 1030 5 630 6 530 1 30anWhen makingfafo ctororganized listding pairspairs, stop finors beginwhen the factto repeat.There could be 1 row of 30 or 30 rows of 1.There could be 2 rows of 15 or 15 rows of 2.There could be 3 rows of 10 or 10 rows of 3.There could be 5 rows of 6 or 6 rows of 5.The factors 5 and 6 are already listed.There are 8 possible arrangements: 1 row of 30, 30 rows of 1, 2 rows of15, 15 rows of 2, 3 rows of 10, 10 rows of 3, 5 rows of 6, or 6 rows of 5.Try ItList the factor pairs of the number.1. 182. 243. 514. WHAT IF? The woodwinds section of the marching band has38 members. Which has more possible arrangements, the brasssection or the woodwinds section? Explain.Key IdeaPrime FactorizationThe prime factorization of a composite number is the number writtenas a product of its prime factors.RememberA prime numberis a whole numbergreater than 1 withexactly two factors,1 and itself. Acomposite number is awhole number greaterthan 1 with factors inaddition to 1 and itself.16Chapter 1ms2019 gr6 Ch01.indb 16You can use factor pairs and a factor tree to help find the primefactorization of a number. The factor tree is complete when only primefactors appear in the product. A factor tree for 60 is shown.60 302 153 560 2 2 3 5, or 2 3 5Numerical Expressions and Factors22Multi-Language Glossary at BigIdeasMath.com1/16/18 4:27 PM

EXAMPLE2 Writing a Prime FactorizationWrite the prime factorization of 48.Choose any factor pair of 48 to begin the factor tree.Tree 1beginningNotice thatnt factorwith differe e sames in thpairs resultrization.prime factoerosite numbEvery compe primehas only onn.factorizatioTree 2Find a factor pair anddraw “branches.”48 242 124 32 248 2 2 3 2 22483Circle the prime factorsas you find them. 162 4Find factors until each branchends at a prime factor.8 2 248 3 2 2 2 2The prime factorization of 48 is 2 2 2 2 3, or 2 3.24Try ItWrite the prime factorization of the number.5. 206. 88Self-Assessment7. 908. 462for Concepts & SkillsSolve each exercise. Then rate your understanding of the success criteriain your journal.WRITING A PRIME FACTORIZATION Write the prime factorization ofthe number.9. 1410. 8611. 4012. 51613. WRITING Explain the difference between prime numbers andcomposite numbers.14.STRUCTURE Your friend lists the following factor pairs andconcludes that there are 6 factor pairs of 12. Explain why your friendis incorrect.1, 122, 63, 412, 16, 24, 315. WHICH ONE DOESN’T BELONG? Which factor pair does not belongwith the other three? Explain your reasoning.2, 284, 146, 9Section 1.3ms2019 gr6 Ch01.indb 177, 8Prime Factorization171/16/18 4:27 PM

EXAMPLE3 Using a Prime FactorizationWhat is the greatest perfect square that is a factor of 1575?Because 1575 has many factors, it is not efficient to list all of its factors andcheck for perfect squares. Use a factor tree to write the prime factorizationof 1575. Then analyze the prime factors to find perfect square factors.15752563 93 31575 3 3 5 5 75 5 7The prime factorization shows that 1575 has three factors other than 1that are perfect squares. 5 5 25(3 5) (3 5) 15 15 2253 3 9So, the greatest perfect square that is a factor of 1575 is 225.Self-Assessmentfor Problem SolvingSolve each exercise. Then rate your understanding of the success criteriain your journal.16. A group of 20 friends plays a cardgame. The game can be played with2 or more teams of equal size. Eachteam must have at least 2 members.List the possible numbers and sizesof teams.h l play.l YouY want eachh row17. You arrange 150 chairs in rows for a schoolto have the same number of chairs. How many possible arrangementsare there? Are all of the possible arrangements appropriate for theplay? Explain.18. What is the least perfect square that is a factor of 4536? What is thegreatest perfect square that is a factor of 4536?19.18Chapter 1ms2019 gr6 Ch01.indb 18DIG DEEPERThe prime factorization of a number is 24 34 54 72.Is the number a perfect square? Explain your reasoning.Numerical Expressions and Factors1/16/18 4:27 PM

1.3PracticeGo to BigIdeasMath.com to getHELP with solving the exercises.Review & RefreshEvaluate the expression.1. 2 42(5 3)(2. 23 4 3252123. 9 5 24 — —)Plot the points in a coordinate plane. Draw a line segment connecting the points.4. (1, 1) and (4, 3)5. (2, 3) and (5, 9)6. (2, 5) and (4, 8)Use the Distributive Property to find the quotient. Justify your answer.7. 408 48. 628 29. 969 3Classify the triangle in as many ways as possible.11.10.12.Concepts, Skills, & Problem SolvingREWRITING A NUMBER Write the number as a product of as many factorsas possible. (See Exploration 1, p. 15.)13. 6014. 6315. 12016. 150FINDING FACTOR PAIRS List the factor pairs of the number.17. 1518. 2219. 3420. 3921. 4522. 5423. 5924. 6125. 10026. 5827. 2528. 7629. 5230. 8831. 7132. 91WRITING A PRIME FACTORIZATION Write the prime factorization ofthe number.33. 1634. 2535. 3036. 2637. 8438. 5439. 6540. 7741. 4642. 3943. 9944. 2445. 31546. 49047. 14048. 640USING A PRIME FACTORIZATION Find the number represented by theprime factorization. 49. 22 32 5 50. 32 52 7 51. 23 112 13Section 1.3ms2019 gr6 Ch01.indb 19Prime Factorization191/16/18 4:27 PM

722The prime factorizationof 72 is 2 2 2 9,362or 21823 9. 52.5YOU BE THE TEACHERYour friend finds theprime factorization of 72. Is your friend correct?Explain your reasoning.9USING A PRIME FACTORIZATION Find the greatest perfect square that is a factor ofthe number.53. 25054. 27555. 39256. 33857. 24458. 65059. 75660. 129061. 220562. 189063. 49564. 472565. VOCABULARY A botanist separates plants into equal groupsof 5 for an experiment. Is the total number of plants in theexperiment prime or composite? Explain.66.REASONING A teacher divides 36 students into equalgroups for a scavenger hunt. Each group should have atleast 4 students but no more than 8 students. What are thepossible group sizes?67. CRITICAL THINKING Is 2 the only even prime number?Explain.68.LOGIC One table at a bake sale has 75 cookies. Anothertable has 60 cupcakes. Which table allows for more rectangulararrangements? Explain.69. PERFECT NUMBERS A perfect number is a number that equals the sum of itsfactors, 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 perfectnumbers between 1 and 27?70.REPEATED REASONING Choose any two perfect squares and find theirproduct. Then multiply your answer by another perfect square. Continue thisprocess. Are any of the products perfect squares? What can you conclude?71.Rectangular Prism72.Volume 40 cubic inches20Chapter 1ms2019 gr6 Ch01.indb 20PROBLEM SOLVING The stage manager of a schoolplay creates a rectangular stage that has whole numberdimensions and an area of 42 square yards. String lights willoutline the stage. What is the least number of yards of stringlights needed to enclose the stage?DIG DEEPERConsider the rectangular prism shown. Usingonly whole number dimensions, how many different prismsare possible? Explain.Numerical Expressions and Factors1/16/18 4:27 PM

30 3 10 There could be 3 rows of 10 or 10 rows of 3. 30 5 6 There could be 5 rows of 6 or 6 rows of 5. 30 6 5 The factors 5 and 6 are already listed. Th ere are 8 possible arrangements: 1 row of 30, 30 rows of 1, 2 rows of 15, 15 rows of 2, 3 rows of 10, 10 rows o