# Divisibility

In number theory, **divisibility** is the ability of a number to evenly divide another number. The study of divisibility resides at the heart of number theory, constituting the backbone to countless fields of mathematics. Within number theory,
the study of arithmetic functions, modular arithmetic, and Diophantine equations all depend on divisibility for rigorous foundation.

A **divisor** of an integer is an integer that can be multiplied by some integer to produce . We may equivalently state that is a **multiple** of , and that is **divisible** or **evenly divisible** by .

## Definition

An integer is divisible by a nonzero integer if there exists some integer such that . We may write this relation as An alternative definition of divisibility is that the fraction is an integer — or using modular arithmetic, that . If does *not* divide , we write that .

### Examples

- divides as , so we may write that .
- divides as , so we may write that .
- The positive divisors of are , , , and .
- By convention, we write that every nonzero integer divides ; so .