How to tell if a number is divisible by 9

9 has the tidiest rule of all of them, and the one with the best party trick attached: keep adding the digits of any multiple of 9 and you always end up at 9 itself.

The rule

A number is divisible by 9 when its digits add up to a multiple of 9.

How to check it

Add all the digits together.
If the total is still big, add its digits again, repeating until one digit is left.
If that final digit is 9, the number divides by 9. Anything else is the remainder.

Worked examples

Is 4620 divisible by 9?

4 + 6 + 2 + 0 = 12

12 is not a multiple of 9. The first one is 9, so no.

Is 59 364 divisible by 9?

5 + 9 + 3 + 6 + 4 = 27 → 2 + 7 = 9

Yes. 59 364 = 9 × 6596.

Is 1629 divisible by 9?

1 + 6 + 2 + 9 = 18 → 1 + 8 = 9

Yes. 1629 = 9 × 181.

Why adding the digits works

Every ten is 9 + 1, every hundred is 99 + 1, every thousand is 999 + 1. The 9s, 99s and 999s are all whole numbers of 9s and can never spoil anything.

What is left from each place is the digit itself, so adding the digits adds up exactly what was left over. The total carries the same remainder as the number it came from.

This is also why the digit sum of any multiple of 9 keeps collapsing back to 9, and why the old bookkeeping check called casting out nines works.

Where it goes wrong

Using the rule for 3 and expecting it to answer 9. A digit sum of 12 passes for 3 and fails for 9.
Stopping at a two-digit total. 5 + 8 + 4 = 17 is not obviously anything; add again to get 8, which settles it.
Thinking the digits have to contain a 9. 1629 has one, 59 364 does not, and both divide by 9.

Questions people ask

What is the digital root?

The single digit you reach by adding digits repeatedly. For multiples of 9 it is always 9, and for everything else it equals the remainder after dividing by 9.

Is every multiple of 9 also a multiple of 3?

Yes, because 9 = 3 × 3. The reverse fails: 12 divides by 3 and not by 9.

Does this work in other number systems?

The same trick works in any base, using one less than the base. In base ten that is 9; in base sixteen it would be 15.