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Lesson 11 Chapter 2 Find Common Factors and Common Multiples 6.NS.B.4 Words to Know greatest common factor least common multiple Real-World Connection Sue and her family are waiting at the airport for a shuttle to take them to their car. They need two shuttles, but both shuttles are leaving as they walk to the shuttle area. One of the shuttles returns to the airport every 9 minutes. The other shuttle returns every 12 minutes. How long will it be before the two shuttles are at the airport at the same time again? Breaking Down the Standard The greatest common factor (GCF) of a set of numbers is the greatest (largest) factor common to all the numbers. The numbers could have several factors in common, or they could have just one. So, to find the GCF, list all of the factors of the numbers. example Find the greatest common factor of 12 and 18. List the factors of each number. • Factors of 12: Write multiplication expressions that have a product of 12: 1 ⫻ 12 List the factors: 1 2⫻6 2 3 4 6 3⫻4 12 • Factors of 18: Write multiplication expressions that have a product of 18: 1 ⫻ 18 2⫻9 3⫻6 1 2 3 6 9 18 Look for factors in each list that are the same, or “common.” 1, 2, 3, 4, 6, 12 1, 2, 3, 6, 9, 18 The greatest common factor is 6. 78 masteryeducation.com Mathematics | Level F Copying is prohibited. List the factors: Find Common Factors and Common Multiples Lesson 11 The distributive property states that for any numbers represented by a, b, and c, a(b ⫹ c) ⫽ ab ⫹ ac. For example, 17(15 ⫹ 29) ⫽ 17 ⫻ 15 ⫹ 17 ⫻ 29. This property and GCF often are used together. example Rewrite 16 ⫹ 24 using the distributive property and GCF. Begin by finding the greatest common factor of 16 and 24. Factors of 16: 1, 2, 4, 8, 16 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The greatest common factor is 8. Use the distributive property to express 16 ⫹ 24 as 8(2 ⫹ 3). Remember, 8(2 ⫹ 3) ⫽ 8 ⫻ 2 ⫹ 8 ⫻ 3 ⫽ 16 ⫹ 24. Also, 8(2 ⫹ 3) ⫽ 8 ⫻ 5 ⫽ 40, so 16 ⫹ 24 ⫽ 40. The least common multiple (LCM) of a set of numbers is the least (smallest) positive multiple common to all the numbers. Every number has an infinite number of multiples. example Find the least common multiple of 4 and 6. Hint, Hint . . . Listing multiples of a number is like skipcounting by that number. Make a list of the first few multiples of each number. • Multiples of 4: 4 8 • Multiples of 6: 6 12 12 You can stop when you find a multiple in common. There are other common multiples, such as 24, 36, and 48, but the smallest, or “least” common multiple is 12. Now solve the Real-World Connection problem. Two shuttles leave the airport at the same time. One of the shuttles returns to the airport every 9 minutes. The other shuttle returns every 12 minutes. How long will it be before the two shuttles are at the airport at the same time again? To solve this problem, find the least common multiple (LCM) of 9 and 12. Multiples of 9: 9, 18, 27, 36 Multiples of 12: 12, 24, 36 Copying is prohibited. The least common multiple is 36. This means that after 36 minutes pass, the shuttles will be at the airport at the same time again. The first shuttle will be returning for the fourth time, and the second shuttle will be returning for the third time. Chapter 2 | Number and Operations masteryeducation.com 79 Lesson 11 Find Common Factors and Common Multiples Guided Practice Hint, Hint . . . 1. 1 When finding possible factors, list them in numerical order. For 48, start at the outside of the list, writing 1 on the left and 48 on the right, and work your way in. Write the factors of 72 directly underneath. Find the greatest common factor of 48 and 72. List the factors of 48. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 List the factors of 72. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Look for the largest factor that occurs in both lists. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 The greatest common factor of 48 and 72 is 24. 2. 48 ⫽ 4 ⫻ 12 72 ⫽ 6 ⫻ 12 You may have noticed that 48 and 72 are multiples of 12. 48 ⫽ 2 ⫻ 2 ⫻ 12 72 ⫽ 3 ⫻ 2 ⫻ 12 4 and 6 share a common factor of 2. 48 ⫽ 2 ⫻ (2 ⫻ 12) ⫽ 2 ⫻ 24 72 ⫽ 3 ⫻ (2 ⫻ 12) ⫽ 3 ⫻ 24 24 is the GCF. Find the least common multiple of 4 and 9. List several multiples of 4. 4, 8, 12, 16, 20, 24, 28, 32, 36 List several multiples of 9. 9, 18, 27, 36 Look for the smallest number that occurs in both lists. 36 is the LCM. Check that your answer makes sense. 4 and 9 have no common factors other than 1, so the first common multiple will be 4 ⫻ 9. 80 masteryeducation.com Mathematics | Level F Copying is prohibited. 4 ⫻ 9 ⫽ 36 Find Common Factors and Common Multiples Lesson 11 Guided Application 1. Select the three correct statements about the factors and/or multiples of the following pairs of numbers. The numbers 8 and 12 have three common factors. B The greatest common factor of 12 and 28 is 2. C The least common multiple of 5 and 25 is 25. D The greatest common factor of 35 and 100 is 5. E Using the distributive property, 12 ⫹ 15 can be rewritten as 3(3 ⫹ 5). A The numbers 8 and 12 have three common factors. Correct. 8 and 12 have common factors of 1, 2, and 4. B The greatest common factor of 12 and 28 is 2. Incorrect. While 2 is a common factor of 12 and 28, the greatest common factor is 4: 4 ⫻ 3 ⫽ 12, 4 ⫻ 7 ⫽ 28. C The least common multiple of 5 and 25 is 25. Correct. 25 and 5 are multiples of 5: 25 ⫽ 5 ⫻ 5, 5 ⫽ 5 ⫻ 1. Sometimes, one number will be a multiple of the other. D The greatest common factor of 35 and 100 is 5. Correct. 35 ⫽ 5 ⫻ 7, and 100 ⫽ 5 ⫻ 20. Since 7 and 20 have no common factors other than 1, 5 is the GCF. E Using the distributive property, 12 ⴙ 15 can be rewritten as 3(3 ⴙ 5). Incorrect. While the greatest common factor is 3, the sum inside the parentheses should be (4 ⫹ 5). The expression should be 3(4 ⫹ 5) because 3 ⫻ 4 ⫽ 12. Copying is prohibited. A Chapter 2 | Number and Operations masteryeducation.com 81 Lesson 11 Find Common Factors and Common Multiples Independent Practice Answer the questions. 1. Hint, Hint . . . 2. Which equation is true and uses the GCF of 12 and 30? Ⓐ 12 ⫹ 30 ⫽ 2(6 ⫹ 15) Ⓑ 12 ⫹ 30 ⫽ 3(4 ⫹ 10) Ⓒ 12 ⫹ 30 ⫽ 6(2 ⫹ 5) Ⓓ 12 ⫹ 30 ⫽ 6(2 ⫹ 7) Use the numbers in the box to write the greatest common factor of each of the following pairs of numbers. The numbers can only be used once. Numbers have a limited (finite) number of factors but an infinite number of multiples. 1 2 3 5 7 9 27 and 36 11 and 14 21 and 35 3. Part A Is 40 the least common multiple of 4 and 10? Write Yes or No in the box. 4. Part B Explain your answer. Copying is prohibited. 82 masteryeducation.com Mathematics | Level F Find Common Factors and Common Multiples 5. Write a number in the box to make the equation true. 45 ⫹ 30 ⫽ 6. (3 ⫹ 2) Circle the number that correctly completes the sentence. The greatest common factor of 24 and 27 is 3 Hint, Hint . . . . Factors will always be less than or equal to the number. Multiples will always be greater than or equal to the number. 4 6 9 7. 8. Lesson 11 Circle the two correct statements. A The least common multiple of 6 and 7 is 21. B The least common multiple of 9 and 5 is 45. C The least common multiple of 3 and 6 is 9. D The least common multiple of 11 and 12 is 1. E The least common multiple of 10 and 8 is 40. Is each equation written as a multiple of a sum of two numbers correctly? Select Yes or No. 9. a. 42 ⫹ 33 ⫽ 3(14 ⫹ 11) Yes No b. 12 ⫹ 9 ⫽ 3(4 ⫹ 3) Yes No c. 26 ⫹ 6 ⫽ 6(4 ⫹ 1) Yes No d. 22 ⫹ 42 ⫽ 22(1 ⫹ 2) Yes No Cindy is having a birthday party, and she wants to make gift bags for her guests. She has 36 game cards and 60 colored pencils to divide among the gift bags. She wants to give the same number of each item to each guest. Write your answers in the boxes. Copying is prohibited. What is the greatest number of gift bags she can make? How many game cards will each guest get? How many colored pencils will each guest get? Chapter 2 | Number and Operations masteryeducation.com 83 Lesson 11 Find Common Factors and Common Multiples 10. Part A What is the greatest common factor of 20 and 16? Write your answer in the box. 11. Part B Explain your answer. Copying is prohibited. 84 masteryeducation.com Mathematics | Level F