Division Tricks
Divisibility by 11 – Alternate the Signs
Elevens hide in a pattern: alternately add and subtract the digits from the right.
This page explains the rule, shows why it works, and walks through worked examples. Then you can try it yourself and check your understanding with practice questions. Use the previous and next links at the bottom to move through the tricks in order.
The rule
The Rule: Add and subtract the digits alternately, from right to left. If the result is 0 or a multiple of 11, the number is divisible by 11.
Why it works
Why It Works: Ten sits just one small step away from eleven, one less than it. That tiny one-step gap means that every time you move one place value to the left — from ones to tens, tens to hundreds, and so on — the effect on the “leftover after dividing by 11” flips sign instead of staying the same. Adding and subtracting the digits alternately is really just tracking that flip, one place value at a time.
Formal Proof (for advanced learners)
Since 10 ≡ −1 (mod 11), it follows that 10^i ≡ (−1)^i (mod 11). Therefore N = ∑ a_i·10^i ≡ ∑ a_i·(−1)^i (mod 11), which is precisely the alternating sum of the digits — adding digits in even positions and subtracting digits in odd positions (or vice versa, depending on how you index from the right).
Examples
- 121 → 1 − 2 + 1 = 0, divisible by 11.
- 2728 → 2 − 7 + 2 − 8 = −11, divisible by 11.
- 3729 → 3 − 7 + 2 − 9 = −11, divisible by 11.
Step by step
Example A: is 9163 divisible by 11?
- Starting from the right, add and subtract the digits alternately: 3 − 6 + 1 − 9 = −11.
- −11 is 0 or a multiple of 11.
- So 9163 is divisible by 11.
Example B: is 5248 divisible by 11?
- Starting from the right, add and subtract the digits alternately: 8 − 4 + 2 − 5 = 1.
- 1 is not 0 or a multiple of 11.
- So 5248 is not divisible by 11.
Try it yourself
Now it is your turn. A new random question appears each time. Choose True or False. If you are right, we show how the trick got you there, step by step. If you are not, you will see the full working two ways, the shortcut and the usual method, so you can spot exactly where your answer differed. Press New Question as often as you like: a few quick rounds is the best way to make the trick feel automatic.
Practice questions
Have a go on paper first, then tap Show answer to check your method.
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Is 55 divisible by 11?
Show answer
Yes. Starting from the right, add and subtract the digits alternately: 5 − 5 = 0. 0 is 0 or a multiple of 11. So 55 is divisible by 11.
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Is 74 divisible by 11?
Show answer
No. Starting from the right, add and subtract the digits alternately: 4 − 7 = −3. −3 is not 0 or a multiple of 11. So 74 is not divisible by 11.
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Is 132 divisible by 11?
Show answer
Yes. Starting from the right, add and subtract the digits alternately: 2 − 3 + 1 = 0. 0 is 0 or a multiple of 11. So 132 is divisible by 11.
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Is 2915 divisible by 11?
Show answer
Yes. Starting from the right, add and subtract the digits alternately: 5 − 1 + 9 − 2 = 11. 11 is 0 or a multiple of 11. So 2915 is divisible by 11.
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Can 8104 players be split equally into 11 teams with none left over?
Show answer
No. Starting from the right, add and subtract the digits alternately: 4 − 0 + 1 − 8 = −3. −3 is not 0 or a multiple of 11. So 8104 is not divisible by 11.
Tips and watch-outs
- A result of 0 counts as divisible: 121 → 1 − 2 + 1 = 0.
- Starting from the left instead only flips the sign of the answer, so it gives the same verdict.