How to prove all numbers which divided by square numbers are also divided by square numbers of prime?

Question

I'm solving algorithm problem. In this problem, there are max and min.

Between max and min, which number is not divided by square number? is problem.

When I solving this problem, many users solved using square numbers of a prime number.

But I don't understand why I can use square numbers of a prime number instead of normal square numbers.

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EX) MAX = 10, MIN = 20

Normal square numbers = $10^2, 11^2 ... 20^2$

Prime square numbers = $11^2, 13^2 ... 19^2$

Users said that I can solve which numbers are not divided by square numbers using prime square numbers.

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Can you explain why it can be?

Answer 1

HINT

By FTA $\forall n\in \mathbb{N}$ we have that $n=\prod p_i^{a_i}$.

Answer 2

If $m^2|n$, take $p$ prime, $p|m$. Then $p^2|m^2$ and $p^2|n$.