Divisibility by 7

How can we tell whether an integer is divisible by 7? Well, it is a bit of a brain teaser, and it is not as simple as the other divisibility tests. There are several methods, and this is probably the simplest. Take the number 574, and suppose we want to know whether it is divisible by 7.

1. First, we take the last digit and double it, that is, 4 · 2 = 8.
2. Now we take all the digits of the original number except the last one, that is, 57, and subtract the previous product from it: 57 − 8 = 49.
3. If this result is divisible by 7, the original number is divisible by 7. Since 49 is divisible by 7, 574 is divisible by 7 too.

If necessary, we can repeat this process several times. For example, is 3409 divisible by 7?

1. In the first step, we double the last digit: 9 · 2 = 18.
2. Now we subtract: 340 − 18 = 322.
3. Since we do not know whether 322 is divisible by 7, we carry on.
4. We double the last digit: 2 · 2 = 4.
5. We subtract it from the rest: 32 − 4 = 28.
6. The number 28 is divisible by 7, so 3409 is divisible by 7 too.