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

1 Answer

May 1, 2016

$856$ 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 $856$ .

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