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# Greatest Common Factors (GCFs)

In math, a factor is a number that, when multiplied by another number or numbers, gets you to the desired number. Usually, numbers can be factored in different combinations. For example, to get to 12, you can multiply $2×6,3×4,4×3,1×12$ .

You could also multiply $2×2×3$ to reach 12, which is called prime factorization. Prime factorization is reducing a number to the smallest whole number factors, which is a unique set of prime numbers the product of which is the desired number.

## What is the greatest common factor of two numbers?

The greatest common factor, or GCF, is the largest whole number that is the factor that is common to two or more given whole numbers. To say it another way, it is the largest number that can be divided evenly into the given numbers.

In order to find the greatest common factor for two numbers, the first step is to find the prime factorization of each of the numbers. Let's start with an example.

Example 1

Find the GCF for 36 and 24.

To find the prime factorization of 36, start by dividing it by 2. $2×18$ . Keep the 2 and divide 18 by 2. Now you have $2×2×9$ . Finally, divide the 9 by 3, and you have $2×2×3×3=36$ .

To find the prime factorization of 24, start by dividing by 2 again. $2×12$ . Keep the 2 and divide 12 by 2. Now you have $2×2×6$ . Divide the 6 by 2 and you have $2×2×2×3$ .

Now look for common factors. Both 36 and 24 have two 2s and one 3 in their prime factorization.

Finally, multiply those common factors to get the GCF of the two numbers. $2×2×3=12$ . The GCF of 36 and 24 is 12.

## What is the greatest common factor of two monomials?

To find the GCF for a pair of monomials, you must also find the prime factorization of the monomials. That means writing out each variable, and if necessary, a -1 to represent a negative number.

Let's try an example.

Example 2

$-20{x}^{2}y{z}^{5}=-1×2×2×5×x×x×y×z×z×z×z×z$

$35{x}^{3}{z}^{2}=5×7×x×x×x×z×z$

The common factors of the two monomials are 5, two x's, and two z's, or

$5{x}^{2}{z}^{2}$

which is the GCF.

## Practice finding the greatest common factor (GCF)

Find the GCF of 26 and 65.

26 – First, divide by 2, you have $2×13$ . Those are both prime numbers, so that is the prime factorization of 26.

65 – First, divide by 5, you have $5×13$ . Those are both prime numbers, so that is the prime factorization of 65.

The common factor is 13, so that is the GCF of 26 and 65.

Find the GCF of 128 and 44.

128 – First, divide by 2, you have $2×64$ . Divide 64 by 2, you have $2×2×32$ . Divide 32 by 2, you have $2×2×2×16$ . Divide 16 by 2, you have $2×2×2×2×8$ . Divide 8 by 2, you have 4. Finally, divide 4 by 2, and you have $2×2×2×2×2×2×2$ . That is the prime factorization of 128.

44 – divide by 2, you have 2 × 22. Divide 22 by 2, you have $2×2×11$ . That is the prime factorization of 44.

The only factors the two prime factorizations have in common are two 2's, so the GCF of 128 and 44 is 4.

## Get help learning about greatest common factors (GCFs)

There is a lot that goes into finding a pair of numbers' greatest common factors. If your student could use some help wrapping their head around the entire process, working with a math tutor would be an excellent solution. A tutor can supplement classroom teaching using methods that are targeted toward your student's learning style, making it more likely your student will understand the information quickly and completely. To learn more about how tutoring can help your student master GCFs, contact Varsity Tutors and speak to one of our Educational Directors today.

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