Divisibility by 11

How can we tell that an integer is divisible by 11? By using a slightly modified sum of the digits. The sum of the digits of a number is simply what you get by adding them up. For example, the sum of the digits of 8,261 is 8 + 2 + 6 + 1 = 17. That sum would help us decide whether a number is divisible by 3, but for divisibility by 11 we have to change it a little.

Instead of adding all the digits, we alternate between adding and subtracting, like this:

$$8-2+6-1=11$$

First we add, then we subtract, and so on. If the result is divisible by 11, the original number is divisible by 11 too. In this case the alternating digit sum is 11, so 8,261 is divisible by 11. Be ready for the sum to come out as zero – in that case the number is divisible by 11. For example, 132 has the sum 1 − 3 + 2 = 0, so it is divisible by 11.

Be ready for the alternating digit sum to come out negative as well. For example, 1,727 has the sum 1 − 7 + 2 − 7 = −11. Since −11 is divisible by 11, so is 1,727.

For completeness, here is a number that is not divisible by 11: take 500. Its alternating digit sum is 5, which is not divisible by 11.