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# Least Common Multiple

The least common multiple (LCM) is the smallest multiple that 2 or more numbers have in common.
There are 2 ways to find the LCM of a group of numbers.

Method 1 is:
1) Write a few multiples of each number
2) The first common number is the LCM
3) If no numbers are common, write a few more multiples of each number
until the LCM is found.

Example: Find the LCM of 30 & 35
Multiples of 30 are: 30, 60, 90, 120, 150, 180, 210
Multiples of 35 are: 35, 70, 105, 140, 175, 210
The LCM is 210.

Method 2 is:
1) Write the prime factorization of each number
2) Underline each different factor once (if a factor occurs more than once for a number, underline the repeating factor each time it occurs in the factorization where it occurs the most)
3) The product of all underlined factors is the LCM

Example: Find the LCM of 30 & 35
The prime factorization of 30 is: 235
The prime factorization of 35 is: 5∙7
The different factors are 2, 3, 5, and 7. 5 is a factor of 30 as well as 35. We use it once since it occurs only once for each number.
The LCM is 2357 = 210

Example: Find the LCM of 20 & 64
The prime factorization of 20 is 2∙2∙5
The prime factorization of 64 is 2∙2∙2∙2∙2∙2
The different factors are 2 and 5. 2 is a factor of 20 as well as 64. We use it 6 times since it occurs most for 64 or 26.
The LCM is 2∙2∙2∙2∙2∙25 = 320.