How to tell if a number is divisible by 12
12 is where the pick-two-factors approach bites back. 12 is 2 × 6 and also 3 × 4, but only one of those pairs gives a working test.
A number is divisible by 12 when it divides by 3 and by 4.
How to check it
- Add the digits. If the total is a multiple of 3, the first half passes.
- Look at the last two digits. If they make a multiple of 4, the second half passes.
- Both have to pass for the number to divide by 12.
Worked examples
Is 4620 divisible by 12?
4 + 6 + 2 + 0 = 12 → multiple of 3 ✓ last two digits 20 = 4 × 5 ✓
Both pass, so yes. 4620 = 12 × 385.
Is 42 816 divisible by 12?
4 + 2 + 8 + 1 + 6 = 21 → multiple of 3 ✓ last two digits 16 = 4 × 4 ✓
Yes. 42 816 = 12 × 3568.
Is 4626 divisible by 12?
4 + 6 + 2 + 6 = 18 → multiple of 3 ✓ last two digits 26 → not a multiple of 4 ✗
No. It divides by 6, but not by 12.
Why 3 and 4, and not 2 and 6
A pair of tests only proves divisibility by their product when the two numbers share no divisor apart from 1. 3 and 4 share nothing but 1, so passing both means the number contains a 3 and two 2s, which is a 12.
2 and 6 overlap: 6 already contains a 2, so passing both only proves the number contains a 6 and at least one 2, which it already had. 18 divides by 2 and by 6 and does not divide by 12.
The same rule of thumb settles other composites: for 15 test 3 and 5, for 20 test 4 and 5, for 24 test 3 and 8.
Where it goes wrong
- Testing 2 and 6. It is the most natural split of 12 and it lets numbers like 18, 30 and 42 through.
- Testing 4 and stopping because the number is obviously even.
- Using the 6 test and assuming 12 follows. Every multiple of 12 divides by 6, but not the other way round.
Questions people ask
What about 24, 36 or 48?
Same method with coprime halves: 24 is 3 and 8, 36 is 4 and 9, 48 is 3 and 16.
Is there a single-step rule for 12?
No, and there does not need to be: two rules a child already knows are faster than any special-purpose procedure.