NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g. to find a common denominator to add two fractions or to find the reduced form of a fraction). California Standards

Vocabulary multiple least common multiple (LCM) The product of any number and a nonzero whole number is a multiple of that number. The common multiple with the least value.

The tires on Kendra’s truck should be rotated every 7,500 miles and 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 nonzero 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 1: Using a List to Find the LCM Find the least common multiple (LCM). A. 2, 7 Multiples of 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. Multiples of 7: 7, 14, 21, 28, 35 Find the least value that is in both lists. The LCM is 14. B. 3, 6, 9 Multiples of 3: 3, 6, 9, 12, 15, 18, 21 List some multiples of each number. Multiples of 6: 6, 12, 18, 24, 30 Multiples of 9: 9, 18, 27, 36, 45 Find the least value that is in all the lists. The LCM is 18.

Example 2A: Using Prime Factorization to Find the LCM Find the least common multiple (LCM). 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 2B: Using Prime Factorization to Find the LCM Find the least common multiple (LCM). 14, 35, 49 Write the prime factorization of each number. 14 = 2  7 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.

Example 3: 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.

Check It Out! Example 1 Find the least common multiple (LCM). A. 3, 7 Multiples of 3: 3, 6, 9, 12, 15, 18, 21 List some multiples of each number. Multiples of 7: 7, 14, 21, 28 Find the least value that is in both lists. The LCM is 21. B. 2, 6, 4 Multiples of 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. Multiples of 6: 6, 12, 18 Multiples of 4: 4, 8, 12 Find the least value that is in all the lists. The LCM is 12.

Find the least common multiple (LCM). Check It Out! Example 2B Find the least common multiple (LCM). 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.

Find the least common multiple (LCM). Check It Out! Example 2A Find the least common multiple (LCM). 50, 110 Write the prime factorization of each number. 50 = 2  5  5 110 = 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.

Check It Out! Example 3 Find the LCM of 24 and 18. 24 = 2  2  2  3 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. 72 hours will elapse before they are over the same location at the same time.