# LCM

LCM stands for Lowest Common Multiple, in other words, the smallest number that is a multiple of each number in a set of numbers.

LCM is helpful for many math operations, including finding a common denominator in fractions. The fastest and most efficient way to find a common denominator is to find the LCM of the denominators.

**How do you find a LCM?**

For many numbers, if you know your times tables, it's easy to figure out the LCM, it might even be obvious.

But for bigger numbers, or if you do not know your times tables, it's helpful to use a system.

*The List System:*

The list system involves listing out the multiples of the numbers you are trying to find the LCM for, and circling lowest multiple that they have in common (to do this, list the multiples more or less simultaneously, so that you don't end up listing many multiples of one number only to find that the first multiple you list for the second number is a multiple in common). To list the multiples, list the number times 1, times 2, times 3, times 4, etc.

*Example: Find the LCM of 15 and 9 *

**15: 15, 30, 45, 60**

**9: 9, 18, 27, 36, 45, 54, 63**

LCM of 15 and 9 =

**15: 15, 30, $\boxed{45}$, 60**

**9: 9, 18, 27, 36, $\boxed{45}$, 54, 63**

*The Prime Factor System:*

Another way to find the LCM is to find the prime factorization of each number. You then make a list of 1) each prime factor that the numbers have in common, and 2) each of the factors that are unique to each number. Multiply all of those numbers together for the LCM.

*Example: Find the GCF of 24 and 36*

$\eqalign{2&4\\/\;&\;\backslash\\6\quad & \quad 4\\/\backslash \quad&\quad /\backslash\\\boxed{2}\quad\boxed{3}\; & \; \boxed{2} \quad \boxed{2}}$

Prime factorization of 24: 2, 2, 2, 3

$\eqalign{3&6\\/\;&\;\backslash\\9\quad & \quad 4\\/\backslash \quad&\quad /\backslash\\\boxed{3}\quad\boxed{3}\; & \; \boxed{2} \quad \boxed{2}}$

Prime factorization of 36: 2, 2, 3, 3

**Prime factors in common: 2, 2, 3**

**Prime factors unique to 24: 2**

**Prime factors unique to 36: 3**

$2 \times 2 \times 3 \times 2 \times 3=72$

**LCM of 24 and 36 is 72.**

No matter what system students use, it's helpful for them to know what a LCM is (and that because it is a multiple, the LCM will never be smaller than the largest number in the set).