Section 3.2 Least Common Multiple 171 Least Common Multiple Section 3.2 1. Least Common Multiple In Section 3.1 we learned how to add and subtract like fractions. To add or subtract fractions with different denominators, we must learn how to convert unlike fractions into like fractions. An essential concept in this process is the idea of a least common multiple of two or more numbers. When we multiply a number by the whole numbers 1, 2, 3, and so on, we form the multiples of the number. For example, some of the multiples of 6 and 9 are shown below. Multiples of 6 Multiples of 9 6 1 6 9 1 9 6 2 12 9 2 18 6 3 18 9 3 27 6 4 24 9 4 36 6 5 30 9 5 45 6 6 36 9 6 54 6 7 42 9 7 63 6 8 48 9 8 72 6 9 54 9 9 81 In red, we have indicated several multiples that are common to both 6 and 9. The least common multiple (LCM) of two given numbers is the smallest whole number that is a multiple of each given number. For example, the LCM of 6 and 9 is 18. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, . . . Multiples of 9: 9, 18, 27, 36, 45, 54, 63, . . . Answers 1. A multiple of a number is the product of the number and a whole number 1 or greater. A factor of a number is a value that divides evenly into the number. 2. 75 3. 60 TIP: There are infinitely many numbers that are common multiples of both 6 and 9. These include 18, 36, 54, 72, and so on. However, 18 is the smallest, and is therefore the least common multiple. If one number is a multiple of another number, then the LCM is the larger of the two numbers. For example, the LCM of 4 and 8 is 8. Multiples of 4: 4, 8, 12, 16, . . . Multiples of 8: 8, 16, 24, 32, . . . Finding the LCM by Listing Multiples Example 1 Find the LCM of the given numbers by listing several multiples of each number. a. 15 and 12 b. 10, 15, and 8 Solution: a. Multiples of 15: 15, 30, 45, 60 Multiples of 12: 12, 24, 36, 48, 60 The LCM of 15 and 12 is 60. b. Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120 Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120 The LCM of 10, 15, and 8 is 120. Concepts 1. Least Common Multiple 2. Finding the LCM by Using Prime Factors 3. Finding the LCM by Using Division by Primes 4. Applications of the LCM 5. Equivalent Fractions and Ordering Fractions Concept Connections 1. Explain the difference between a multiple of a number and a factor of a number. Skill Practice Find the LCM by listing several multiples of each number. 2. 15 and 25 3. 4, 6, and 10
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