# Divisibility by 11

Two-digit numbers that are divisible by 11 are hard to miss because they simply repeat the same digit twice. Here are all the numbers less than 100 that are divisible by 11:

11 22 33 44 55 66 77 88 99

For numbers between 100 and 200, use this rule: Every three-digit number whose first and third digits add up to its second digit is divisible by 11. For example, suppose you want to decide whether the number 154 is divisible by 11. Just add the first and third digits:

1 + 4 = 5

Because these two numbers add up to the second digit, 5, the number 154 is divisible by 11. If you divide, you get 154 ÷ 11 = 14, a whole number.

Now suppose you want to figure out whether 136 is divisible by 11. Add the first and third digits:

1 + 6 = 7

Because the first and third digits add up to 7 instead of 3, the number 136 isn’t divisible by 11. You can find that 136 ÷ 11 = 12r4.

For numbers of any length, the rule is slightly more complicated, but it’s still often easier than doing long division. To find out when a number is divisible by 11, place plus and minus signs alternatively in front of every digit, then calculate the result. If this result is divisible by 11 (including 0), the number is divisible by 11; otherwise, the number isn’t divisible by 11.

For example, suppose you want to discover whether the number 15,983 is divisible by 11. To start out, place plus and minus signs in front of alternate digits (every other digit):

+1 − 5 + 9 − 8 + 3 = 0

Because the result is 0, the number 15,983 is divisible by 11. If you check the division, 15,983 ÷ 11 = 1,453.

Now suppose you want to find out whether 9,181,909 is divisible by 11. Again, place plus and minus signs in front of alternate digits and calculate the result:

+9 − 1 + 8 − 1 + 9 − 0 + 9 = 33

Because 33 is divisible by 11, the number 9,181,909 is also divisible by 11. The actual answer is

9,181,909 ÷ 11 = 834,719