Digital roots and casting out nines

Computing a digital root

There is also a formula: for any positive whole number n, the digital root is 1 + ((n − 1) mod 9). In words, it is the remainder after dividing by 9, except that multiples of 9 give 9 rather than 0.

Why digit sums reveal remainders

The reason is that 10 leaves remainder 1 when divided by 9, and so do 100, 1,000 and every other power of 10. That means:

A number always leaves the same remainder mod 9 as the sum of its digits. Repeating the digit sum does not change the remainder, which is why the digital root ends up equal to it.

Casting out nines

Because digital roots behave well under addition and multiplication, they give a quick check for arithmetic. To check 347 × 86 = 29,842:

1. Digital root of 347: 3 + 4 + 7 = 14 → 5.
2. Digital root of 86: 8 + 6 = 14 → 5.
3. Multiply the roots: 5 × 5 = 25 → 7.
4. Digital root of 29,842: 2 + 9 + 8 + 4 + 2 = 25 → 7. The check passes.

Bookkeepers used this method for centuries, long before calculators. Its name comes from the habit of crossing out any digits that add up to 9, since they do not affect the result.

What the check cannot catch

Casting out nines only compares remainders, so errors that leave the remainder unchanged slip through. Swapping two digits is the classic example: 29,842 and 29,482 have the same digital root. Misplacing a zero, or an error of exactly 9, 18 or 27, will also pass. A passed check means “probably right”, not “certainly right”.

Divisibility tests from the same idea

Since a number and its digit sum leave the same remainder after dividing by 9, a number is divisible by 9 exactly when its digit sum is. The same argument works for 3, because 10 also leaves remainder 1 when divided by 3. So 6174 is divisible by both 3 and 9, and 2026 is divisible by neither.

A related test works for 11. Because 10 leaves remainder −1 when divided by 11, you alternate the signs of the digits from the right: for 6174 that gives 4 − 7 + 1 − 6 = −8, which is not a multiple of 11, so 6174 is not divisible by 11. For 1089 it gives 9 − 8 + 0 − 1 = 0, and indeed 1089 = 11 × 99.

The link with Kaprekar’s routine

Rearranging digits does not change their sum, so the descending and ascending arrangements in Kaprekar’s routine always have the same remainder mod 9. Their difference therefore has remainder 0: it is a multiple of 9. That is why every step of the routine produces a multiple of 9.

The same idea explains the 1089 trick and much of the structure in why every number reaches 6174.