If $p \mid a^n$ then does $p^n \mid a^n$? I'm trying to figure out if the statement is true or not and I need to prove it if so.
Let $p$ be a  prime and $a$ be an integer.
If $p\mid a^n$ , is it true that $p^n\mid a^n$ ?
I'm not sure how i would approach this problem.
I started as $a^n = p \cdot b$ for some $b \in \mathbb{Z}$.

This can be done using a basic property.
Note that if $p|xy$ then $p|x$ or $p|y$.
So $p|a\cdot a^{n-1}$ means $p|a$ (when we are pretty much done) or $p|a^{n-1}$.
So there is a simple induction to show $p|a$ and we are done as before
