---
tag: 00J8E3HX0M
kind: concept
name: Divisibility rule
aliases: [divisibility-test]
msc: [11A63]
sources:
  - url: https://en.wikipedia.org/wiki/Divisibility_rule
    locator: lead section
same_as:
  wikidata: Q858520
  wikipedia: 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|0TPW25CBN6]] 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|0R9D4CEJEB]], [[divisibility by 9|0ESHD7ZQDX]] and [[divisibility by 11|09YBMJTNBM]].
[[Casting out nines|0E7W0EM6J6]] 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 $11$.