Question

How can you use prime factorization to determine if 856 is evenly divisible by 7?

Answer 1

`856` is not evenly divisible by `7`

Explanation:

You can reduce the size of the problem if you can separate out other prime factors.

`color(white)(00000)856`
`color(white)(00000)"/"color(white)(0)"\"`
`color(white)(0000)2color(white)(00)428`
`color(white)(0000000)"/"color(white)(0)"\"`
`color(white)(000000)2color(white)(00)214`
`color(white)(000000000)"/"color(white)(0)"\"`
`color(white)(00000000)2color(white)(00)107`

If we knew that `107` is prime then we would be done.

• It is not divisible by `2` since the last digit is not even.
• It is not divisible by `3` since the sum of the digits `1+0+7=8` is not divisible by `3`.
• It is not divisible by `5` since the last digit is neither `5` nor `0`.

The next factor to try would be `7`, which is the one we're wanting to test anyway.

I'm not sure this factorisation has helped us much, but let us at least split it down to make the arithmetic a little easier:

`107 / 7 = (70+35+2) / 7 = 70/7+35/7+2/7 = 10+5+2/7 = 15 2/7`

So `107` is not divisible by `7` and neither is `856`.