# Least Common Multiple

## What Is Least Common Multiple?

In Factorization, the **Least Common Multiple** (**LCM**)
is also known as **Least Common Denominator** (**LCD**).
So what does it mean? The simple answer to this
question is find the smallest number that can be divided evenly by both numbers.
Example: the Least Common Multiple of 108 and 126 is 756: 756 /
108 = 7 and 756 / 126 = 6.

To find the least common multiple, factor both numbers into their primes using Prime Factorization. Each prime factor of either number must appear in the least common multiple at least as many times as it does in each one. Before going further, here is recommend reading: Exponents, and Prime Factorization.

Let's find the least common multiple of 108 and 126:

- Prime Factorization of 108 = 2 x 2 x 3 x 3 x 3 =
2
^{2}x 3^{3}

- Prime Factorization of 126 = 2 x 3 x 3 x 7 = 2 x
3
^{2}x 7

- The LCM is 2 x 2 x 3 x
3 x 3 x 7 = 2
^{2}x 3^{3}x 7 = 756.

How do we get the answer 756? The answer is to take the greatest value of each term of the Prime Factorization of 108 and 126 and multiply them together.

- Thus 2
^{2}is greater than 2^{1}, 3^{3}is greater than 3^{2}, and 7 is included because it is unique. Multiply these numbers together to get the answer 2^{2}x 3^{3}x 7 = 756. The product is the GCF / GCD 756.

(a) 756 ÷ 108 = 7 and

(b) 756 ÷ 126 = 6 and

(c) note both divide evenly which is our goal.

- If you already know the GCF
of the two numbers: LCM = (product of the two
numbers) / Greatest Common Factor e.g. example -> LCM = (Number1
x Number2) / GCF ⇒ (108 × 126) ÷ 18 = 756.

There is a important exception. See Relative Primes.