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

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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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