Common Factor

Pronunciation: /ˈkɒm ən ˈfæk tər/ ?

A common factor in a number or expression that is a factor of two or more values.[1] A factor can be a number or an expression.

Take, for example, the two numbers 4 and 6. Both numbers are divisible by 2, so 2 is a common factor of 4 and 6.

Finding Common Factors of Numbers

Example of a prime factor tree
Figure 1: Prime factor tree

As you get more experience working with numbers, you will often be able to find common factors by inspection. This means you can look at the two numbers and figure out quickly what is the common factor.

If you don't have enough experience yet to find common factors by inspection, you can use prime factorization to find common factors. The method most often used to get the prime factorization of a number is a factor tree. Figure 1 is an example of a factor tree.

Example 1

What are the common factors of 30 and 45?

Step 1: Factor 30 and 45.

factor tree of 30Factor tree of 45
Figure 2: Factor tree of 30Figure 3: Factor tree of 45

Step 2: Find the common prime factors.

factor tree of 30 with 3 and 5 highlightedFactor tree of 45 with 3 and 5 highlighted.
Figure 4: Factor tree of 30.Figure 5: Factor tree of 45.

Step 3: Calculate all common factors.

The common factors are 3, 5, 3·5=15.

Expressions as Common Factors

An expression can also be a common factor. For example, the expression x-2 is a factor of the expression (x-2)(x+1). Since (x-2)(x+1) = x2-x-2, the expression x-2 is also a factor of the polynomial x2-x-2.

Example 2

What are the common factors, if any, of (x-2)(x+1)(x+4) and (x-4)(x+1)(x+4)? Since (x+1) and (x+4) are factors of the two expressions and are common (meaning the same), (x+1) and (x+4) are common factors of (x-2)(x+1)(x+4) and (x-4)(x+1)(x+4).

More Information

  • McAdams, David. Greatest Common Factor. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Greatest%20Common%20Factor.
  • McAdams, David. Prime Factorization. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Prime%20Factorization.

Cite this article as:


Common Factor. 2010-02-01. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/commonfactor.html.

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Revision History


2010-02-01: Added "References" (McAdams, David.)
2009-01-20: Added information on expressions as common factors (McAdams, David.)
2008-07-07: Added example (McAdams, David.)
2008-06-07: Corrected spelling (McAdams, David.)
2008-03-25: Revised More Information to match current standard (McAdams, David.)
2008-02-05: Fixed missing vocabulary link (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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