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EXAMPLE 1 Finding the Least Common Multiple A veterinarian at an animal clinic is on call every four days. Today is Saturday, and the veterinarian is on call. In how many more days will the veterinarian be on call on a Saturday again? Animal Clinic SOLUTION The veterinarian is on call every 4 days. A Saturday occurs every 7 days. Find the least common multiple of 4 and 7.

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EXAMPLE 1 Finding the Least Common Multiple Use a calendar. Start with a Saturday and circle every 4 days on a calendar. You can see that the veterinarian will be on call on a Saturday in 28 days. METHOD 1

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EXAMPLE 1 Finding the Least Common Multiple Make a list. List the multiples of each number. METHOD 2 Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32,... Multiples of 7 : 7, 14, 21, 28, 35, 42, 49, 56,... The least common multiple is the first number that appears in both lists. The LCM of 4 and 7 is 28.

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EXAMPLE 1 Finding the Least Common Multiple METHOD 3 Use prime factorization. Write the prime factorization of each number. Circle the greatest power of each prime factor. The LCM of 4 and 7 is the product of the circled numbers: 4 = 2 2 7= 7 4 = 2 2 7= 7 2 2 7 = 28

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EXAMPLE 1 Finding the Least Common Multiple In 28 days, the veterinarian will be on call on a Saturday. ANSWER

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EXAMPLE 2 Finding the Least Common Multiple Find the LCM of 32, 96, and 120 using prime factorization. SOLUTION Write the prime factorization of each number and circle the greatest power of each prime factor. STEP 1 32 = 2 5 96 = 2 5 3 120 = 2 3 3 5

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EXAMPLE 2 Finding the Least Common Multiple STEP 2 Find the product of the circled factors. 2 5 3 5 The LCM of 32, 96, and 120 is 480. ANSWER = 480

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EXAMPLE 3 Finding the LCM of Monomials Find the LCM of 6x 2 y and 9x 4 z. SOLUTION Factor each expression using exponents and circle the greatest power of each factor. STEP 1 Find the product of the circled factors. STEP 2 6x 2 y = 2 3 x 2 y 9x 4 z = 3 2 x 4 z 2 3 2 x 4 y z = 18x 4 yz

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EXAMPLE 3 Finding the LCM of Monomials The LCM of 6x 2 y and 9x 4 z is 18x 4 yz. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 1. What If? In Example 1, suppose the veterinarian is on call every 3 days. In how many more days will the vet be on call on a Saturday again? In 21 days, the veterinarian will be on call on a Saturday. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 Find the least common multiple of the numbers. 2. 6, 15

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 6 and 15 is 30. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 3. 4, 20

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 4 and 20 is 20 ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 4. 12, 28

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 12 and 28 is 84. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 5. 24, 36, and 72

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 24, 36 and 72 is 72. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 6. 8x 3, 20x 7

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 8x 3 and 20x 7 is 40x 7. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 7. 12y 4, 36y 8

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 12y 4 and 36y 8 is 36y 8. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 8. 4ab 2, 10a 2 b

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 4b 2 and 10a 2 b is 20a 2 b 2. ANSWER

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GUIDED PRACTICE for Examples 1,2 and 3 9. 6m 3 np 2, 8mp 3

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GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 6m 3 np 2 and 8mp 3 is 24m 3 np 3. ANSWER

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