Decimal digit-sum divisibility test ($a\equiv {\sum }_i{a}_i(\mathrm{mod}d)(d\mid 9)$)

For a nonnegative integer with decimal expansion $a={\sum }_i{a}_{i}10^i$, the congruence $10\equiv 1(\mathrm{mod}d)$ gives $a\equiv {\sum }_i{a}_i(\mathrm{mod}d)$ whenever $d\mid 9$. Thus divisibility by three or nine is equivalent to divisibility of the sum of its decimal digits by the same number. More generally a base-$b$ expansion has the analogous rule for every divisor of $b-1$. This follows by repeated addition and multiplication in modular arithmetic; it does not require an empirical rule about decimal notation.