GCF and LCM of monomials

Why you need them

With whole numbers, the least common multiple helps you add fractions and the greatest common factor helps you simplify them. Algebra uses both in exactly the same way: the GCF of monomials is the first step in factoring, and the LCM gives you the common denominator of rational expressions.

A quick refresher with numbers. Take \(24 = 2^3 \cdot 3\) and \(36 = 2^2 \cdot 3^2\):

• the GCF uses the shared prime factors with the smaller exponent: \(2^2 \cdot 3 = 12\);
• the LCM uses every prime factor with the larger exponent: \(2^3 \cdot 3^2 = 72\).

Variables behave just like prime factors.

Greatest common factor

You can check it: dividing each monomial by \(10x^2y\) gives a monomial every time (\(2x^2y\), \(3y^4z\) and \(x\)). That's what "common factor" means.

Least common multiple

Practice