Divisibility rule

Summary

A divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$. It uses the digits of $n$, and it does not need a full division. Most rules for base $10$ come from the congruences of the powers $10^k$ modulo $d$. For example, $10 \equiv 1 \pmod 3$, so $n$ and the sum of its digits have the same remainder modulo $3$.

Remarks

For example, $2$ divides $n$ if and only if the last digit of $n$ is even. Also $5$ divides $n$ if and only if the last digit is $0$ or $5$. See divisibility by 3, divisibility by 9 and divisibility by 11. Casting out nines uses the rule for $9$ to check computations.

Explanations

L1 human

You can see at once that $1{,}234{,}570$ is a multiple of $10$, because it ends with $0$. Divisibility rules give similar quick tests for other divisors, for example $3$, $4$, $9$ and $11$.