Every integer is a square times a square-free integer

Builds on Square-free integer Prime factorization

Statement

Let $n$ be a positive integer. Then there are positive integers $a$ and $b$ such that $n = a^2 b$ and $b$ is square-free.

Remarks

The integers $a$ and $b$ are unique. The number $b$ is the product of the primes $p$ for which the exponent of $p$ in $n$ is odd. For example, $72 = 2^3 \cdot 3^2 = 6^2 \cdot 2$ , so $a = 6$ and $b = 2$ . The proof uses the prime factorization of $n$ . Write each exponent $e$ as $e = 2 q + r$ with $r \in \{0, 1\}$ .

Reading path

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Divisibility definition
Prime number definition
Prime factorization definition
Square-free integer definition
Every integer is a square times a square-free integer statement

Tag	`080Q3RPKY6`
URI	`m2://080Q3RPKY6`
Kind	statement, proposition
Writing	W0 stub
MSC	11A51
Status	This copy of the site has no build data. Run `m2 status 080Q3RPKY6`.
Uses	Square-free integer, Prime factorization

Every integer is a square times a square-free integer

Statement

Remarks

Reading path

Further reading

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