Warm Up Write the prime factorization of each number. 1. 68 2. 225 3. 940 Find the greatest common factor. 4. 27 and 45 5. 32 and 80 6. 50 and 71 22 · 17 32 · 52 22 · 5 · 47 9 16 1

Learn to find the least common multiple of two or more whole numbers.

Insert Lesson Title Here Vocabulary multiple least common multiple (LCM)

The maintenance schedule on Ken’s pickup truck shows that the tires should be rotated every 7,500 miles and that the oil filter should be replaced every 5,000 miles. What is the lowest mileage at which both services are due at the same time? To find the answer you can use least common multiples.

A multiple of a number is a product of that number and a whole number A multiple of a number is a product of that number and a whole number. Some multiples of 7,500 and 5,000 are as follows: 7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . . 5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000, . . . A common multiple of two or more numbers is a number that is a multiple of each of the given numbers. So 15,000 and 30,000 are common multiples of 7,500 and 5,000.

The least common multiple (LCM) of two or more numbers is the common multiple with the least value. The LCM of 7,500 and 5,000 is 15,000. This is the lowest mileage at which both services are due at the same time.

Example : Using a List to Find the LCM Find the least common multiple (LCM). A. 2, 7 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. 7: 7, 14, 21, 28, 35 Find the least value that is in both lists. The LCM is 14. B. 3, 6, 9 3: 3, 6, 9, 12, 15, 18, 21 List some multiples of each number. 6: 6, 12, 18, 24, 30 9: 9, 18, 27, 36, 45 Find the least value that is in all the lists. The LCM is 18.

Insert Lesson Title Here Try This: Example Find the least common multiple (LCM). A. 3, 7 List some multiples of each number. 3: 3, 6, 9, 12, 15, 18, 21 7: 7, 14, 21, 28 Find the least value that is in both lists. The LCM is 21. B. 2, 6, 4 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. 6: 6, 12, 18 4: 4, 8, 12 Find the least value that is in all the lists. The LCM is 12.

Example : Using Prime Factorization to Find the LCM Find the least common multiple (LCM). A. 60, 130 60 = 2 · 2 · 3 · 5 Write the prime factorization of each number. 130 = 2 · 5 · 13 Circle the common prime factors. 2, 2, 3, 5, 13 List the prime factors, using the circled factors only once. 2 · 2 · 3 · 5 · 13 Multiply the factors in the list. The LCM is 780.

Example : Using Prime Factorization to Find the LCM Find the least common multiple (LCM). B. 14, 35, 49 14 = 2 · 7 Write the prime factorization of each number. 35 = 5 · 7 Circle the common prime factors. 49 = 7 · 7 List the prime factors, using the circled factors only once. 2, 5, 7, 7 2 · 5 · 7 · 7 Multiply the factors in the list. The LCM is 490.

Insert Lesson Title Here Try This: Example 2A Find the least common multiple (LCM). A. 50, 130 50 = 2 · 5 · 5 Write the prime factorization of each number. 110 = 2 · 5 · 11 Circle the common prime factors. List the prime factors, using the circled factors only once. 2, 5, 5, 11 2 · 5 · 5 · 11 Multiply the factors in the list. The LCM is 550.

Insert Lesson Title Here Try This: Example B. 18, 36, 54 18 = 2 · 3 · 3 Write the prime factorization of each number. 36 = 2 · 2 · 3 · 3 Circle the common prime factors. 54 = 2 · 3 · 3 · 3 List the prime factors, using the circled factors only once. 2, 2, 3, 3, 3 2 · 2 · 3 · 3 · 3 Multiply the factors in the list. The LCM is 108.

Additional Example : Application Mr. Washington will set up the band chairs all in rows of 6 or all in rows of 8. What is the least number of chairs he will set up? Find the LCM of 6 and 8. 6 = 2 · 3 8 = 2 · 2 · 2 The LCM is 2 · 2 · 2 · 3 = 24. He will set up at least 24 chairs.

Insert Lesson Title Here Try This: Example Two satellites are put into orbit over the same location at the same time. One orbits the earth every 24 hours, while the second completes an orbit every 18 hours. How much time will elapse before they are once again over the same location at the same time? Find the LCM of 24 and 18. 24 = 2 · 2 · 2 · 3 18 = 2 · 3 · 3 The LCM is 2 · 2 · 2 · 3 · 3 = 72.