Greatest Common Factor

Pronunciation: /greɪt.est ˈkɒm.ən ˈfæk.tər/ Explain

A greatest common factor is the largest number or expression that is a factor of two or more numbers or expressions. A greatest common factor can also be called a greatest common divisor. In British English, the greatest common factor is called the highest common factor and the greatest common divisor is called the highest common divisor. The abbreviations for these terms are:

Term	Abbreviation
greatest common factor	gcf
greatest common divisor	gcd
highest common factor	hcf
highest common divisor	hcd
Table 1: Abbreviations

The greatest common factor of 8 and 12 is 4. The greatest common factor of (x + 3)(x - 2) and (2x - 6)(x - 2) is (x - 2).

How to Find a Greatest Common Factor Using Prime Factorization

There are two steps to finding a greatest common factor of numbers or expressions:

1. Find all prime factors of each of the numbers or expressions.
2. Identify every prime factor that is common to all of the numbers or expressions.

Example: Find the greatest common factor of 60 and 70.

1. The prime factorization of 60 is 2 · 2 · 3 · 5. The prime factorization of 70 is 2 · 5 · 7.
2. The common factors are 2 and 5. The greatest common factor is 2 · 5 = 10.

Understanding Check

For each step, find the answer, then click on Click for answer.

1. What is the greatest common factor of 12 and 6?
   • What are the prime factors of 12?
     Click for answer The prime factors of 12 are 2, 2 and 3. This can also be written 2² · 3 = 12.
   • What are the prime factors of 6?
     Click for answer The prime factors of 6 are 2 and 3. This can also be written 2 · 3 = 6.
   • What prime factors are common to 6 and 12?
     Click for answer The common prime factors of 6 and 12 are 2 and 3.
   • Multiply the prime factors together to get the greatest common factor.
     Click for answer 2 · 3 = 6. The greatest common factor of 6 and 12 is 6.
2. What is the greatest common factor of 12 and 6?
   • What are the prime factors of 12?
     Click for answer The prime factors of 12 are 2, 2 and 3. This can also be written 2² · 3 = 12.
   • What are the prime factors of 6?
     Click for answer The prime factors of 6 are 2 and 3. This can also be written 2 · 3 = 6.
   • What prime factors are common to 6 and 12?
     Click for answer The common prime factors of 6 and 12 are 2 and 3.
   • Multiply the prime factors together to get the greatest common factor.
     Click for answer 2 · 3 = 6. The greatest common factor of 6 and 12 is 6.
3. What is the greatest common factor of 9 and 12?
   • What are the prime factors of 9?
     Click for answer The prime factors of 9 are 3 and 3. This can also be written 3² = 9.
   • What are the prime factors of 12?
     Click for answer The prime factors of 12 are 2, 2 and 3. This can also be written 2² · 3 = 12.
   • What prime factors are common to 9 and 12?
     Click for answer The common factor of 9 and 12 is 3.
   • Since there is only one common prime factor, the greatest common factor is the prime factor.
     Click for answer The greatest common factor of 9 and 12 is 3.

   How to Use the Euclidean Algorithm to Find Greatest Common Factors

   The Euclidean algorithm is a quick method of finding the greatest common factor of two numbers. It is based on the fact that the greatest common factor of two numbers is the same as the greatest common factor of either number and their difference. For example, use 51 and 85. gcf(51,85) = 17. Now subtract 51 from 85. 85 - 51 = 34. gcf(85,34) = 17 and gcf(51,34) = 17.

   The Euclidean algorithm uses this fact to find the greatest common factor. To find the greatest common factor using the Euclidean algorithm:

   Iteration	First Number	Second Number	Difference	Discussion
   1	124	217	217 - 124 = 93	Since 193 and 124 are the smaller of the three numbers, use them for the next step.
   2	124	93	124 - 93 = 31	Since 93 and 31 are the smaller of the three numbers, use them for the next step.
   3	93	31	93 - 31 = 62	Since 62 and 31 are the smaller of the three numbers, use them for the next step.
   4	62	31	62 - 31 = 31	Since 31 and 31 are the smaller of the three numbers, use them for the next step.
   5	31	31	31 - 31 = 0	Since the difference is zero, the greatest common factor of 217 and 124 is 31.
   Table 2

   References

   1. McAdams, David E.. All Math Words Dictionary, greatest common factor. 2nd Classroom edition 20150108-4799968. pg 87. Life is a Story Problem LLC. January 8, 2015. Buy the book

   More Information

   • McAdams, David E.. Common. allmathwords.org. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/c/common.html.
   • Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.

   Cite this article as:

   McAdams, David E. Greatest Common Factor. 4/21/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/greatestcommonfactor.html.

   Image Credits

   • All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

   Revision History

   4/21/2019: Updated equations and expressions to new format. (McAdams, David E.)
   12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
   8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
   7/10/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
   1/3/2011: Added highest common factor, highest common divisor and abbreviations. (McAdams, David E.)
   2/8/2010: Added "References". (McAdams, David E.)
   8/7/2008: Added Understanding Check. (McAdams, David E.)
   8/5/2008: Added How To. (McAdams, David E.)
   6/7/2008: Corrected spelling. (McAdams, David E.)
   7/12/2007: Initial version. (McAdams, David E.)