Modular Arithmetic proof? Every integer n can be written as $n=a^2 b$, where $b$ is a product of primes

Question

Every positive integer $k$ can be written as $k = m^2 n$ (where $m$ and $n$ are positive integers) and $n$ is a product of distinct primes that could be empty)

Answer 1

Probably the easiest way to see this is by induction. It's true for $1$; assume it's true for all numbers below $k$.

If $k$ is not divisible by the square of a prime, then it must be a product of different primes, so $k=1^2k$ works. Otherwise $p^2$ divides $k$ for some prime $p$. But then $k=p^2j$ for some $j<k$, and by assumption $j=a^2b$ where $b$ is a product of primes. Then $k=(ap)^2b$.

Answer 2

For $k=1$, choose $m=n=1$

Otherwise $$k=p_1^{a_1}\cdots p_n^{a_n}$$

with primes $p_1,\cdots p_n$ and positive integers $a_1\cdots a_n$

Collect the primes with odd exponent and let $P$ be the product of the primes. If no such prime exists , set $P=1$. Then, $\frac{k}{P}$ is a perfect square and we have the desired representation.