Suppose $p$ is a prime number and $a$ is an integer. Show that if $p \mid a^n$, then $p^n \mid a^n$ for any $n \geq 1$?

Question

I know that if $p \mid a^n$, I can say $a^n = pr$ for some integer $r$, you can also conclude that $\gcd(p, a^n) = p$, but I'm not sure how to use that information if I even can to show that $p^n \mid a^n$.

Answer 1

Hint: What is $\gcd(p,a)$? $1$ or $p$?

Answer 2

You probably already have the fact that $p$ is prime and $p\mid ab$ implies that $p\mid a$ or $p\mid b$. You can use this to "descend" from $p\mid a^n$ to $p\mid a^{n-1}$ or $p\mid a$, then $p\mid a^{n-2}$ or $p\mid a$ etc., until you have $p\mid a$.