# How do you find the LCM of monomials?

## Question:

How do you find the LCM of monomials?

## Multiples:

A multiple of a number is a product that is calculated by multiplying the given number by a whole number. For example, the multiples of {eq}3 {/eq} are {eq}3, 6, 9, 12, ... {/eq}. When working with algebraic expressions containing variables, a multiple of an expression can be calculated by multiplying by a variable also, so {eq}4x^2 {/eq} is considered a multiple of {eq}2x {/eq} because {eq}2x*2x=4x^2 {/eq}. In a set of numbers, the least common multiple (LCM) is the smallest value that is a multiple of all numbers in the set.

To find the LCM of monomials, find the LCM of the coefficients, then multiply each variable factor the greatest number of times it occurs in the set of numbers.

For example, find the LCM of {eq}12x^2y^4z^5 {/eq} and {eq}15x^3y^3z^2 {/eq}.

First, find the LCM of the coefficients by writing the prime factorization:

{eq}12=2*2*3 {/eq}

{eq}15=3*5 {/eq}

LCM of the coefficients is {eq}2^2*3*5=60 {/eq}

For the variables, the LCM will contain the the greatest number of times each variable occurs, so {eq}x {/eq} occurs {eq}3 {/eq} times at most in {eq}15x^3y^3z^2 {/eq}, {eq}y {/eq} occurs {eq}4 {/eq} times at most in {eq}12x^2y^4z^5 {/eq}, and {eq}z {/eq} occurs {eq}5 {/eq} times at most in {eq}12x^2y^4z^5 {/eq}.

The LCM of {eq}12x^2y^4z^5 {/eq} and {eq}15x^3y^3z^2 {/eq} is {eq}60x^3y^4z^5 {/eq}