How to Tell Whether a Number Is Divisible by 11

How do you tell whether a number is divisible by 11?

To test whether a number divides by 11, tag its digits plus and minus alternately, starting with plus on the right, and add them up. If the total is a multiple of 11, including 0, the number divides by 11. For 2728, the total is 8 − 2 + 7 − 2 = 11, and 2728 = 11 × 248. A negative total counts too: 5093 gives 3 − 9 + 0 − 5, which is minus 11, and 5093 = 11 × 463.

It works because 10 is one less than 11, so each place value leaves a remainder of plus or minus 1 against 11, and the sign flips at every step to the left.

When the total is not a multiple of 11, it still gives the remainder once you take away 11s: 2729 gives 9 − 2 + 7 − 2 = 12, and 12 − 11 = 1, so 2729 leaves remainder 1.

What does the test look like on a longer number?

Tag the digits + and − from the right and add. For 918082: 2 − 8 + 0 − 8 + 1 − 9 = −22, a multiple of 11, so it divides: 918082 = 11 × 83462. For 2013: 3 − 1 + 0 − 2 = 0, and 0 counts, so 2013 = 11 × 183. A total that is not a multiple of 11, such as 5 or −7, means the number does not divide.

Why does the sign flip at every place?

10 is one less than 11, so 10 leaves remainder −1 when divided by 11, and 100 = 10 × 10 leaves (−1) × (−1) = +1. Each place multiplies by another 10, so the remainders run +1, −1, +1, −1 from the right. A number and its alternating digit sum therefore leave the same remainder against 11, and a remainder of 0 means it divides.