---
tag: 096PW7Q014
kind: definition
name: Divisibility
aliases: [divides, divisor, factor, multiple]
msc: [11A05]
defines: 0PG2H7WKGG
sources:
  - url: https://en.wikipedia.org/wiki/Divisor
    locator: Definition
---

## Definition

Let $a$ and $b$ be integers.
The integer $a$ divides $b$ when there is an integer $k$ such that $b = a k$.
The notation $a \mid b$ means that $a$ divides $b$, and $a \nmid b$ means that it does not.
When $a \mid b$, $a$ is a divisor (or a factor) of $b$, and that $b$ is a multiple of $a$.

## Remarks

Each integer divides $0$, because $0 = a \cdot 0$. The integer $0$ divides only $0$.
The integers $1$ and $-1$ divide each integer.
If $a \mid b$ and $b \mid c$, then $a \mid c$. On the natural numbers, divisibility is a partial order.
If $a \mid b$ and $a \mid c$, then $a \mid (x b + y c)$ for all integers $x$ and $y$.

## Explanations

### L1 human

The integer $a$ divides $b$ when $b$ is in the list $\dots, -2a, -a, 0, a, 2a, \dots$ of the multiples of $a$.
For example, $3 \mid 12$ because $12 = 3 \cdot 4$.
But $5 \nmid 12$, because no integer $k$ gives $5 k = 12$.