Division Tricks
Divisibility by 6 – Pass Both the 2 and 3 Tests
Six is 2 × 3, so a number must pass both of the easy tests you already know.
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: A number is divisible by 6 if it is divisible by both 2 and 3.
Why it works
Why It Works: 6 is built from two smaller building blocks, 2 and 3, which share no common factors of their own. If a number passes both the “is it even” test and the “digit sum” test, it already contains a hidden factor of 2 and a hidden factor of 3 — and multiplying those two together guarantees a factor of 6 sitting inside the number.
Formal Proof (for advanced learners)
6 = 2·3, and gcd(2, 3) = 1 (they share no common factors). If 2 | N and 3 | N, then because 2 and 3 are coprime, their product also divides N, i.e. 6 | N. So N is divisible by 6 if and only if it is divisible by both 2 and 3.
Examples
- 54 is even, and 5 + 4 = 9 (divisible by 3). So 54 is divisible by 6.
- 36 is even, and 3 + 6 = 9 (divisible by 3). So 36 is divisible by 6.
- 27 is odd, so it fails the first test and is not divisible by 6.
Step by step
Example A: is 8412 divisible by 6?
- Test for 2: the last digit is 2, which is even.
- Test for 3: 8 + 4 + 1 + 2 = 15, which is divisible by 3.
- So 8412 is divisible by 6.
Example B: is 1238 divisible by 6?
- Test for 2: the last digit is 8, which is even.
- Test for 3: 1 + 2 + 3 + 8 = 14, which is not divisible by 3.
- So 1238 is not divisible by 6.
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 72 divisible by 6?
Show answer
Yes. Test for 2: the last digit is 2, which is even. Test for 3: 7 + 2 = 9, which is divisible by 3. So 72 is divisible by 6.
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Is 45 divisible by 6?
Show answer
No. Test for 2: the last digit is 5, which is odd. Test for 3: 4 + 5 = 9, which is divisible by 3. So 45 is not divisible by 6.
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Is 130 divisible by 6?
Show answer
No. Test for 2: the last digit is 0, which is even. Test for 3: 1 + 3 + 0 = 4, which is not divisible by 3. So 130 is not divisible by 6.
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Is 516 divisible by 6?
Show answer
Yes. Test for 2: the last digit is 6, which is even. Test for 3: 5 + 1 + 6 = 12, which is divisible by 3. So 516 is divisible by 6.
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Can 2004 cards be dealt equally to 6 players with none left over?
Show answer
Yes. Test for 2: the last digit is 4, which is even. Test for 3: 2 + 0 + 0 + 4 = 6, which is divisible by 3. So 2004 is divisible by 6.
Tips and watch-outs
- Failing either test means “no”: 21 is odd, so it fails the test for 2 even though its digits add to 3.
- This “both tests” idea only works when the two numbers share no factor: 18 passes the tests for 2 and 6 but is not divisible by 12.