Question

What is the GCF of 60 and 45?

Answer 1

`15`

Explanation:

Here are two methods:

Method 1 - Identify common prime factors and multiply

Find the prime factorisations of each of the two numbers.

We can use factor trees if that helps:

`color(white)(000000)60`
`color(white)(00000)"/"color(white)(00)"\"`
`color(white)(0000)2color(white)(000)30`
`color(white)(0000000)"/"color(white)(00)"\"`
`color(white)(000000)2color(white)(000)15`
`color(white)(000000000)"/"color(white)(00)"\"`
`color(white)(00000000)3color(white)(0000)5`

`color(white)(000000)45`
`color(white)(00000)"/"color(white)(00)"\"`
`color(white)(0000)3color(white)(000)15`
`color(white)(0000000)"/"color(white)(00)"\"`
`color(white)(000000)3color(white)(0000)5`

So:

`60 = 2*2*color(blue)(3)*color(blue)(5)`

`45 = 3*color(blue)(3)*color(blue)(5)`

So the common prime factors are `3` and `5` with multiplicity `1`

Hence the GCF is:

`3*5 = 15`

`color(white)()`
Method 2 - Quotient and remainder

Given two numbers proceed as follows:

• Divide the larger number by the smaller to give a quotient and a remainder.

• If the remainder is `0` then the smaller number is the GCF.

• Otherwise repeat with the smaller number and the remainder.

In our example:

`60 / 45 = 1" "` with remainder `15`

`45 / 15 = 3" "` with remainder `0`

So the GCF is `15`.