Question

How can you simplify the following fraction using prime factorization?: `456/939`

Answer 1

`456/939 = 152/313`

Explanation:

We can factor `456` as expressed by this factor tree:

`color(white)(00000)456`
`color(white)(0000)"/"color(white)(000)"\"`
`color(white)(000)2color(white)(0000)228`
`color(white)(0000000)"/"color(white)(000)"\"`
`color(white)(000000)2color(white)(0000)114`
`color(white)(0000000000)"/"color(white)(000)"\"`
`color(white)(000000000)2color(white)(0000)57`
`color(white)(0000000000000)"/"color(white)(00)"\"`
`color(white)(000000000000)3color(white)(000)19`

We can factor `939` as expressed by this factor tree:

`color(white)(00000)939`
`color(white)(0000)"/"color(white)(000)"\"`
`color(white)(000)3color(white)(0000)313`

It might take us a while to find that `313` is prime, but we don't actually need to. The only factors we need to check for are those common with `456`, i.e. `2`, `3` and `19`.

So the only common factor is `3` (once) and we find:

`456/939 = (color(red)(cancel(color(black)(3)))xx152)/(color(red)(cancel(color(black)(3)))xx313) = 152/313`

`color(white)()`
Alternative method

Rather than doing all of this factoring, we can find the GCF of `456` and `939` using the following method:

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

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

• Otherwise repeat with the smaller number and the remainder.

So in our example:

`939/456 = 2` with remainder `27`

`456/27 = 16` with remainder `24`

`27/24 = 1` with remainder `3`

`24/3 = 8` with remainder `0`

So the GCF is `3` and we can write:

`456/939 = ((456/3))/((939/3)) = 152/313`