---
tag: 004JW7A2N5
kind: definition
name: Square-free integer
aliases: [squarefree]
msc: [11A51]
defines: 08X0VCR2NW
uses: [0SN0MAEKYX, 096PW7Q014]
sources:
  - url: https://en.wikipedia.org/wiki/Square-free_integer
    locator: lead section
---

## Definition

A nonzero integer $n$ is square-free when no square of a [[prime number|0SN0MAEKYX]] divides $n$.
That is, there is no prime $p$ such that $p^2 \mid n$.

## Remarks

Equivalently, $n$ is square-free when the only perfect squares that divide $n$ are $1$.
A positive integer is square-free if and only if each exponent in its [[prime factorization|0AEKDFCYFA]] is $1$.
The first square-free positive integers are $1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15$. The integer $12 = 2^2 \cdot 3$ is not square-free.
The integer $0$ is not square-free, because each square divides $0$.

## Explanations

### L1 human

A number is square-free when no prime appears twice in its factorization.
For example, $30 = 2 \cdot 3 \cdot 5$ is square-free, but $18 = 2 \cdot 3 \cdot 3$ is not, because $9$ divides $18$.