# What is the greatest common factor? How do you know when you have found the greatest one?

A common factor of two or more terms divides each of the terms. The greatest common factor is the largest of the common factors.

To find the greatest common factor, begin by writing each term as a product of prime factors. Then the greatest common factor is the product of each prime factor that appears in the factorizations of each term to the lowest power.

Ex. The gcf(18,24):

`18=2*3^2`

`24=2^3*3`

Since 2 and 3 appear in both factorizations, we take the product of the lowest power of 2 and 3 that appear. The greatest common factor is 2*3=6.

Ex. The gcf of `36x^2y^5,48x^3y^2,60x^5yz` :

`36x^2y^5=2^2* 3^2*x^2*y^5`

` ` `48x^3y^2=2^4*3*x^3*y^2`

`60x^5yz=2^2*3*5*x^5*y*z`

The following all appear as factors in each factorization:2,3,x,y. So the greatest common factor is `2^2*3*x^2*y` or `12x^2y`

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In Arithmetic, Greatest Common Factor (GCF) of two or more numbers is the GREATEST number that these numbers are divisible by.

For example, suppose you need to find the GCF of 12 and 18.

All factors of 12 are 1, 2, 3, 4, 6, and 12.

All factors of 18 are 1, 2, 3, 6, 9, and 18.

As you can see, 2, 3, and 6 are COMMON factors of 12 and 18, but 6 is the GREATEST one, so GCF = 6.

The formal way to find GCF (and make sure you have found the greatest one) is by writing each number as a product of prime factors (prime factorization):

In the example above,

12 = 2 * 2* 3

18 = 2 * 3 * 3

GCF will be the product of ALL COMMON prime factors:

GCF = 2 * 3 = 6

Visit these websites for more details and examples:

http://www.ck12.org/arithmetic/Greatest-Common-Factor-Using-Lists/lesson/Greatest-Common-Factor-Using-Lists/

http://www.ck12.org/arithmetic/Greatest-Common-Factor-Using-Factor-Trees/

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