Digital root (${\mathrm{dr}}_b(n)$)

Repeatedly sum the numerical digits of a nonnegative integer in base $b$ until a single numerical digit remains. For $n>0$, the result is $1+((n-1)\mathrm{mod}(b-1))$, while the result for zero is zero. This follows because $b\equiv 1(\mathrm{mod}b-1)$, so summing numerical digits preserves the integer congruence class modulo $b-1$.