Least Common Multiple
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Least Common Multiple
Lesson 3-6
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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, ____
24 30
3 15 18
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, 80If 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, 36Answer: 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, 25 beats 22.)
• 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 = 23 x 11102 = 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. 23 beats regular 2.
LCM: 23 x 3 x 11 x 17 = 4,488
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Homework Time