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Least Common Multiple

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Multiples A multiple is formed by multiplying a given number by the counting numbers. The counting numbers are 1, 2, 3, 4, 5, 6, etc.

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Example: List the multiples of 4: 4 x 1 = 4 4 x 2 = 8 4 x 3 = 12 4 x 4 = 16 4 x 5 = 20 4 x 6 = 24 Counting Numbers So, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, etc.

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What are the first five multiples of 13? 13 x 1 =13 13 x 2 = 26 13 x 3 = 39 13 x 4 = 52 13 x 5 = 65 13, 26, 39, 52, 65

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Find the Missing Multiples 6, 12, 18, ____, ____ ___, 6, 9, 12, ____, ____, 21 ___, 24, 36, 48, 60, ____ 2430 31518 12 72

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Least Common Multiple (LCM) The least common multiple is the smallest number that is common between two lists of multiples.

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EXAMPLE: Find the LCM of 12 and 18 The multiples of 12: 12 x 1 = 12 12 x 2 =24 12 x 3 = 36 12 x 4 = 48 12 x 5 =60 The multiples of 18: 18 x 1 = 18 18 x 2 = 36 18 x 3 = 54 18 x 4 = 72 18 x 5 = 90

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12, 24, 36, 48, 60 18, 36, 54, 72, 90 The first number you see in both lists is 36. The least common multiple of 12 and 18 is 36.

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Example 2: Find the LCM of 9 and 10 9, 18, 27, 36, 45, 54, 63, 72 10, 20, 30, 40, 50, 60, 70, 80 If you don’t see a common multiple, make each list go further. 81, 90, 99 90, 100, 110 The LCM of 9 and 10 is 90

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Example 3: Find the LCM of 4 and 12 4, 8, 12, 16 12, 24, 36 Answer: 12

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Using Prime Factorization to find the LCM Make factor trees for each number. Write the prime factorization in exponent form. Identify all of the prime numbers among the prime factorizations. Every prime number will be used. When the same prime number occurs in more than one prime factorization, select the prime that there’s more of (for example, 2 5 beats 2 2.) Multiply your prime numbers to get the LCM.

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Example: Find the LCM of 88 and 102 88 2 x 44 2 x 22 2 x 11 102 2 x 51 3 x 17 88 = 2 3 x 11 102 = 2 x 3 x 17 We have to use every prime number. The numbers we will use are 2, 3, 11, and 17. We have 2 in both places. Select the “bigger” one. 2 3 beats regular 2. LCM: 2 3 x 3 x 11 x 17 = 4,488

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Homework Time

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