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QUESTION

Prove that, for any prime $p$, if $p|a^2n$ then $p|a$ and hence
$p^n|a^n$.



ANSWER

Suppose $a$ is written as a product of prime powers as
$p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$. Then
$a^n=p_1^{n\alpha_1}p_2^{n\alpha_2}\ldots p_k^{n\alpha_k}$. Now
$p|a^n$, so $p$ must be equal to one of the primes $p_1,p_2\ldots
p_k$. Thus $p|a$ and $p^n|a^n$, as required.





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