Digital Root Calculator

The single digit you get by repeatedly summing the digits of a non-negative integer until only one remains.

Because 10 is congruent to 1 mod 9, every power of 10 is also congruent to 1 mod 9. So the sum of the digits of n is congruent to n mod 9, and after enough iterations you reach a single digit equal to n mod 9 (with 9 substituted when n mod 9 equals 0).

Yes - it is a quick divisibility-by-9 check and a special case of digital invariants. It is also a classical pre-calculator way to spot arithmetic errors (casting out nines).

Additive persistence is how many times you must sum a number's digits before reaching a single digit. For example 12345 → 15 → 6 takes two steps, so its additive persistence is 2 (the final single digit is the digital root).

A number's digital root equals the number itself modulo 9 (with 9 used instead of 0). So the digital root is 9 exactly when the number is divisible by 9, which is why casting out nines works as an arithmetic check.