Division Tricks
Divisibility by 3 – Add the Digits
Big numbers can be checked for 3 in seconds: just add up their digits.
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: If the sum of all the digits of a number is divisible by 3, then the number itself is divisible by 3.
Why it works
Why It Works: Every place value in our number system — 1, 10, 100, 1000 — is just one more than a multiple of 3 (1, 9+1, 99+1, 999+1 …). So when a digit sits in the hundreds place, say, almost all of its value is already a multiple of 3, and only a tiny leftover of “1 per digit” escapes. Adding up the digits is really just collecting all those small leftovers — and if they add up to a multiple of 3, the whole number must be too.
Formal Proof (for advanced learners)
Write N = ∑ a_i·10^i, where a_i are the digits of N. Since 10 ≡ 1 (mod 3), it follows that 10^i ≡ 1 (mod 3) for every power i. Therefore N ≡ ∑ a_i·1 = ∑ a_i (mod 3): N and its digit sum always leave the same remainder when divided by 3, so one is divisible by 3 exactly when the other is.
Examples
- 123 → 1 + 2 + 3 = 6, divisible by 3.
- 453 → 4 + 5 + 3 = 12, divisible by 3.
- 217 → 2 + 1 + 7 = 10, not divisible by 3.
Step by step
Example A: is 8712 divisible by 3?
- Add the digits: 8 + 7 + 1 + 2 = 18.
- 18 is divisible by 3.
- So 8712 is divisible by 3.
Example B: is 5473 divisible by 3?
- Add the digits: 5 + 4 + 7 + 3 = 19.
- 19 is not divisible by 3.
- So 5473 is not divisible by 3.
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 45 divisible by 3?
Show answer
Yes. Add the digits: 4 + 5 = 9. 9 is divisible by 3. So 45 is divisible by 3.
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Is 82 divisible by 3?
Show answer
No. Add the digits: 8 + 2 = 10. 10 is not divisible by 3. So 82 is not divisible by 3.
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Is 351 divisible by 3?
Show answer
Yes. Add the digits: 3 + 5 + 1 = 9. 9 is divisible by 3. So 351 is divisible by 3.
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Is 1004 divisible by 3?
Show answer
No. Add the digits: 1 + 0 + 0 + 4 = 5. 5 is not divisible by 3. So 1004 is not divisible by 3.
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Can 7392 marbles be shared equally among 3 friends with none left over?
Show answer
Yes. Add the digits: 7 + 3 + 9 + 2 = 21. 21 is divisible by 3. So 7392 is divisible by 3.
Tips and watch-outs
- Add all the digits, not just some of them.
- If the sum is still large, add its digits again: 6789 → 30 → 3, so 6789 is divisible by 3.