How to tell if a number is divisible by 6
6 has no rule of its own. It borrows two, and the mistake almost everyone makes is checking one of them, finding it works, and stopping there.
A number is divisible by 6 when it is even and its digits add up to a multiple of 3, both at once.
How to check it
- Check the last digit. If it is 0, 2, 4, 6 or 8, the number passes the test for 2.
- Add the digits. If the total is a multiple of 3, the number passes the test for 3.
- Only if both are true does the number divide by 6.
Worked examples
Is 4620 divisible by 6?
ends in 0 → even ✓ 4 + 6 + 2 + 0 = 12 → multiple of 3 ✓
Both tests pass, so yes. 4620 = 6 × 770.
Is 1155 divisible by 6?
ends in 5 → not even ✗ 1 + 1 + 5 + 5 = 12 → multiple of 3 ✓
It passes the 3 test but fails the 2 test, so no.
Is 4622 divisible by 6?
ends in 2 → even ✓ 4 + 6 + 2 + 2 = 14 → not a multiple of 3 ✗
Even, but the digit sum fails, so no.
Why both tests, and why these two
6 = 2 × 3, and 2 and 3 share no divisor apart from 1. A number that contains a 2 and a 3 among its factors therefore contains a 6.
Passing only one test proves only half of it. 1155 has a 3 in it but no 2, so the best it can do is an odd multiple of 3. 4622 has a 2 but no 3.
The same logic is why the test for 12 is 3 and 4 rather than 2 and 6: the two halves have to be coprime, or they overlap and let impostors through.
Where it goes wrong
- Checking that the number is even and stopping. That is the single most common error with 6.
- Testing 2 and 3 on different numbers by accident. Both tests apply to the original number, not to a half of it.
- Assuming every even multiple of 3 needs checking twice. It does: being even and having a digit sum of 12 are two separate facts.
Questions people ask
Could I halve the number and test for 3 instead?
Yes, that works too: if a number is even and its half divides by 3, it divides by 6. The two-test version is usually faster because the digit sum does not change much.
Is every multiple of 6 also a multiple of 3?
Yes, and of 2. The multiples of 6 are exactly the numbers that appear in both the 2 and 3 times tables.