What are the divisibility rules for the number seven?

Answer 1

Divisibility by 7 can be tested by taking rightmost digit of the number, and subtracting twice it's value from the rest of the number. If the result is divisible by seven, then the original number is as well. That pattern can be repeated on the resulting number until we get down to 7, -7, or 0.

For example, take the number 9268. We'll take the last digit, "8", double it, and subtract from the first three, "926". 926 - (2 * 8) = 910. Not sure? Repeat the process. 91 - (2 * 0) = 91, and 9 - (2 * 1) = 7. Seven is of course divisible by itself, so we know then that 9268 is as divisible by seven as well.

This works because of a relationship between the digits, and a particular multiple of 7. First, consider our original number, 9268. We'll break that down into the two numbers we're working with, 926 and 8. Let's take two variables, A and B. we'll say that A equals 926 and B equals 8. In that case we can say:

9268 = 10A + B

In that case, if 9268 is divisible by 7, then 10A + B is as well. We also know that if any number is divisible by seven, then any products of that number are divisible by seven as well. This means that if 10A + B is divisible by 7, then 2(10A + B), or 20A + 2B is as well.

At this point, we know that 20A + 2B has a factor of 7, but we don't know if either A or B does. We do however know that 21A would have a factor of 7, no matter what the value of A is. We know this because 7 is a factor of 21. With that in mind, we can say:

if 10A + B is divisible by 7, then 10A + 2B - 21A is also divisible by 7.

and then we can simplify it:
10A + 2B - 21A = -A + 2B

Whether the number is positive or negative does not affect it's divisibility, so we can easily say that if -A + 2B is divisible by 7, then A - 2B is also. Recall that B is our rightmost digit, and A is the rest of the number. This is why this method of testing works.