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A divisibility rule tests whether one integer divides another by replacing it with a simpler equivalent condition. For a base-$b$ expansion $N={\sum }_i{d}_i{b}^i$, the congruence class of $b$ modulo the candidate divisor determines a rule.

In base ten, $10\equiv 1(\mathrm{mod}9)$, so $N$ and its digit sum have the same residue modulo nine. Also $10\equiv -1(\mathrm{mod}11)$, so divisibility by eleven is equivalent to divisibility of the alternating digit sum. For example, $121$ passes because $1-2+1=0$. These tests follow from modular arithmetic and depend on the numerical base.