# The Greatest Common Factor

## ...of Two Numbers:

The greatest common factor is the largest whole number that is a factor of the two given whole numbers.  In other words, it is the largest number that can be divided evenly into the two given numbers.

To find the GCF of two whole numbers, find the prime factorization of each, and then find the product of all common factors .

Example 1:

Find the GCF of $60$ and $42$ .

First, find the prime factorizations.

$\begin{array}{l}60={2}\cdot 2\cdot {3}\cdot 5\\ 42={2}\cdot {3}\cdot 7\end{array}$

Common factors are shown in red. Their product is:

$2\cdot 3=6$

So, the GCF is $6$ .

## ...of Two Monomials:

In the prime factorization of a monomial, include all the variables (and a $-1$ factor if necessary), and proceed as above.

Example 2:

Find the GCF of:

$-27{p}^{2}q{r}^{5}$

and

$15{p}^{3}{r}^{3}$

First, find the prime factorization of each monomial.

$\begin{array}{l}-27{p}^{2}q{r}^{5}=-1\cdot {3}\cdot 3\cdot 3\cdot {p}\cdot {p}\cdot q\cdot {r}\cdot {r}\cdot {r}\cdot r\cdot r\\ 15{p}^{3}{r}^{3}={3}\cdot 5\cdot {p}\cdot {p}\cdot p\cdot {r}\cdot {r}\cdot {r}\end{array}$

Common factors are shown in red. Their product is:

$3\cdot p\cdot p\cdot r\cdot r\cdot r$

So, the GCF is $3{p}^{2}{r}^{3}$ .