# Finding the Greatest Common Factor

**Definition:** The Greatest Common Factor (GCF) is the largest
number/expression that `

divides into two or more expressions evenly .

**For Example:** For the numbers 18 and 27, 3 is a common factor, but 9 is
the greatest

common factor , since 9 is the largest number that divides into 18 and 27

evenly.

**Finding the GCF: **One approach to finding the GCF is looking at the
prime factors that

occurs the least (look for the smallest exponent ) in each of the numbers or
expressions

that are involved. For instance, in the previous example , 18 and 27, factor each
number

into its prime factors.

The least exponent on the 3 is two and on the 2 is zero (since 27 does not
have any

factors of 2) so the GCF is **3 ^{2} = 9.**

Another example of finding the GCF of 90 and 120:

The least exponent of each factor is one so the GCF is 2●3●5 = 30.

**Examples for Finding the GCF of Algebraic Expressions :**

The same approach is used to find the GCF of algebraic expressions — factor
into prime

factors first .

**Example:** Find the GCF of 12x^{2}y^{3}w and 20xy^{2}.

Choose the least exponent for each factor. So the GCF is 2^{2}●x●y^{2}
(3, 5 or w did not occur

in both expressions so they are not part of the GCF).

**Example:** Find the GCF of 3x^{3} + 6x^{2}
and 6x^{2} – 24

The GCF is 3(x + 2)

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