How do you find the GCF of 60r^2, 45r^3?

Apr 25, 2017

$15 {r}^{2}$

Explanation:

First, let's identify the factors of $60$ and $45$

For $60$ we have...

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

and for $45$ we have

1, 3, 5, 9, 15, 45

For our Greatest Common Factor or GCF, we need to find the biggest number that is a factor of both $60$ and $45$

In this case that is $15$

60 = 1, 2, 3, 4, 5, 6, 10, 12, $\textcolor{red}{15}$, 20, 30, 60

45 = 1, 3, 5, 9, $\textcolor{red}{15}$, 45

So now we can say we have

$\textcolor{red}{15} {r}^{2}$ and $\textcolor{red}{15} {r}^{3}$

but remember that we can simplify the exponents of the variables as well.

We have ${r}^{2}$ and ${r}^{3}$

What is the largest exponent that these two have in common?
Think of it like this...

${r}^{2}$ is $r$ and ${r}^{2}$

${r}^{3}$ is $r$, ${r}^{2}$ and ${r}^{3}$

See how we count up to find the exponent of greatest power
So, now we can see that the GCF for the $r$ is ${r}^{2}$

${r}^{2}$ is $r$ and $\textcolor{g r e e n}{{r}^{2}}$

${r}^{3}$ is $r$, $\textcolor{g r e e n}{{r}^{2}}$ and ${r}^{3}$

So now we know that our GCF is:

$\textcolor{red}{15} \textcolor{g r e e n}{{r}^{2}}$