# 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: __2__∙__3__∙__5__

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 __2__∙__3__∙__5__∙__7__ = 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 2^{6}.

The **LCM** is __2∙2∙2∙2∙2∙2__∙__5__ = 320.