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.

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

EXAMPLE 1 Finding the Least Common Multiple METHOD 2 Make a list. List the multiples of each number. 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.

EXAMPLE 1 Finding the Least Common Multiple METHOD 3 Use prime factorization. Write the prime factorization of each number. 4 = 22 7 = 7 Circle the greatest power of each prime factor. 4 = 22 7 = 7 The LCM of 4 and 7 is the product of the circled numbers: 22 7 = 28

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

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 = 25 96 = 25 3 120 = 23 3 5

EXAMPLE 2 Finding the Least Common Multiple STEP 2 Find the product of the circled factors. 25 3 5 = 480 The LCM of 32, 96, and 120 is 480. ANSWER

EXAMPLE 3 Finding the LCM of Monomials Find the LCM of 6x2y and 9x4z. SOLUTION Factor each expression using exponents and circle the greatest power of each factor. STEP 1 6x2y = 2 3 x2 y 9x4z = 32 x4 z Find the product of the circled factors. STEP 2 2 32 x4 y z = 18x4yz

EXAMPLE 3 Finding the LCM of Monomials The LCM of 6x2y and 9x4z is 18x4yz. ANSWER

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

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

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

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

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

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

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

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

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

GUIDED PRACTICE for Examples 1,2 and 3 6. 8x3, 20x7

GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 8x3 and 20x7 is 40x7. ANSWER

GUIDED PRACTICE for Examples 1,2 and 3 7. 12y4, 36y8

GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 12y4 and 36y8 is 36y8. ANSWER

GUIDED PRACTICE for Examples 1,2 and 3 8. 4ab2, 10a2b

GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 4b2 and 10a2b is 20a2b2. ANSWER

GUIDED PRACTICE for Examples 1,2 and 3 9. 6m3np2, 8mp3

GUIDED PRACTICE for Examples 1,2 and 3 The LCM of 6m3np2 and 8mp3 is 24m3np3. ANSWER