How to Find a Remainder Mod 9 from the Digit Sum

Updated 4 October 2026

A number's remainder mod 9 is the same as its digit sum's remainder.

4573 mod 9 = 1

How do you find the remainder when a number is divided by 9?

To find the remainder when a number is divided by 9, add its digits, and keep adding until one digit is left. For 4573, the digits give 4 + 5 + 7 + 3 = 19, then 1 + 9 = 10, then 1 + 0 = 1, so 4573 leaves remainder 1. It works because 10 leaves remainder 1 when divided by 9, and so do 100 and 1000, since each is one more than 9, 99 or 999. So every digit counts only its own value, and a number and its digit sum leave the same remainder. If the digits fold down to 9, the remainder is 0: 738 gives 7 + 3 + 8 = 18, then 1 + 8 = 9, and 738 = 9 × 82. You can also cross out any 9s, or any digits that add to 9, before adding: in 4573 the 4 and the 5 go, which leaves 7 + 3 = 10.

1. Every power of ten leaves remainder 1 against 9, so each digit counts itself.

2. Add the digits: 19. Still two digits, so fold again: 1 + 9 = 10, then 1.

3. That last single digit is the remainder: 4573 leaves 1 when divided by 9.

All 36 shortcuts: the Magic Tricks library.