Divisibility of eleven

How do we find out that a whole number is divisible by eleven? By using a slightly modified digit sum. By digit sum, we normally mean the sum of the digits of a number. For example, the number 8261 has a digit sum equal to 8 + 2 + 6 + 1 = 17. This digit sum would help us determine if the number is divisible by three, but we still need to adjust the sum to make it divisible by eleven.

Instead of adding the digits all the time, we will alternate adding and subtracting as follows:

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

First we subtract, then we add, and so on. In this case, the adjusted digit sum came out to be 11, so the number 8261 is divisible by 11. Be prepared that the sum may come out to be zero, in which case the number is divisible by 11. For the example, the number 132 has a sum of 1 − 3 + 2 = 0 and is therefore divisible by 11.

Similarly, be prepared that the adjusted digit sum may come out negative, for example 1727 has a sum of 1 − 7 + 2 − 7 = −11. However, the number -11 is divisible by 11 and therefore 1727 is divisible by 11.

For completeness, let us add an example of a number that is not divisible by 11, for example the number 500, whose adjusted digit sum is 5, which is not divisible by 11.