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Greatest Common Factor in Algebra (Mathematics Glossary)

What Is the Greatest Common Factor in Algebra?

The greatest common factor in algebra is the largest factor that is common to two or more terms.

Understanding the Greatest Common Factor in Algebra

A factor is a quantity that divides exactly into a term.

It is one of the numbers, letters and brackets (or a product of them) that are multipled together to make a term.

For example,

2xy = 2 times x times y
2xz = 2 times x times z

The greatest common factor is the largest collection of numbers and letters that make up both terms.

In our example, 2x is the greatest common factor.

2x is the greatest common factor of 2xy and 2xz

2x is the largest term that divides exactly into both original terms (2xy and 2xz).

2xy ÷ 2x = y

2xz ÷ 2x = z

How to Find the Greatest Common Factor in Algebra

The greatest common factor of x2(y + 1) and x(y + 1)2z can be found.

FInd the factors of both terms...

factors of x squared y plus 1 and x y plus 1 squared z

...then find the largest collection of factors that appears in both terms:

greatest common factors of x squared y plus 1 and x y plus 1 squared z

The greatest common factor is x(y + 1).

Read more about how to find the greatest common factor in algebra

Curriculum
Algebra Lessons
Interactive Test
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Here's a second test on the greatest common factor.
Here's a third test on the greatest common factor.

Note

What's in a Name?

The Greatest Common Factor (GCF) is so called because it is the largest (greatest) factor that is the same (common) to the numbers.

It may sometimes be called:

  • Greatest Common Divisor (GCD)

  • Highest Common Factor (HCF)

Finding the Greatest Common Factor of More than Two Terms

In the examples here, the greatest common factor has been been found for two terms. But the same procedure extends to any number of terms.

Why Is the Greatest Common Factor in Algebra Useful?

The Greatest Common Factor in algebra is useful when: