Finding the Greatest Common Factor (GCF)

After studying this lesson, you will be able to:

• Find the prime factorization of an integer.
• Find the GCF for a set of monomials.

Prime numbers are those numbers greater thanone whose only factors are one and itself. A partial list ofprime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

One way to find a prime factorization of an integer is to use"factor trees".

Example 1

Find the prime factorization of 15

Factor the number down until all you have left is primenumbers

Write the prime factorization: 3 · 5

Example 2

Find the prime factorization of 30

Factor the number down until all you have left is primenumbers

Write the prime factorization: 2 · 3 · 5

Example 3

Find the prime factorization of -525

Factor out the negative to begin with

Write the prime factorization: -1 · 3 · 5 ²· 7

Example 4

Find the prime factorization of 20a ² b

Don't worry about the variables until the last step.

Write the prime factorization: 2 ² · 5 · a · a· b (we factored the variables, too)

Example 5

Find the prime factorization of 60a ² b ²

Don't worry about the variables until the last step.

Write the prime factorization: 2 ² · 3 · 5 · a· a · b · b

Greatest Common Factors

Sometimes we need to be able to find the greatest commonfactor of a set of numbers. The greatest common, or GCF, is thelargest number that will divide evenly into each of the numbersin a set.

Example 6

Find the GCF for the set of numbers: 10, 12, 20

The largest number that will go into each of these numbers is2.

Example 7

Find the GCF for the set of numbers: 6, 18, 36

The largest number that will go into each of these numbers is6.

Example 8

Find the GCF for the set of numbers: 4, 8, 10

The largest number that will go into each of these numbers is2.

Example 9

Find the GCF for the set of numbers: 8a ² b, 18a ²b ² c

The first thing we do is find the GCF for the coefficients -just like we've been doing. The largest number that will go intoeach of the coefficients is 2.

Since we have variables, we have to find their GCF also. For avariable to be included in the GCF, each term must have thevariable. If the variable is in each term, we take the lowestexponent of the variable and include it in the GCF.

In this case, both terms have a and both terms have b . Wewill include a ² because that is the lowest power of a. We will include b because that is the lowest power of b .

The GCF will be 2 a ² b

Example 10

Find the GCF for the set of numbers: 3x ² y, 12x ⁴y ², 9x ² y

The first thing we do is find the GCF for the coefficients.The largest number that will go into each of the coefficients is3.

Since we have variables, we have to find their GCF also. For avariable to be included in the GCF, each term must have thevariable. If the variable is in each term, we take the lowestexponent of the variable and include it in the GCF.

In this case, both terms have x and both terms have y . Wewill include x ² because that is the lowest power of x. We will include y because that is the lowest power of y .

The GCF will be 3x ² y