Prime Divides Power

Theorem
Let $p$ be a prime number.

Let $a, n \in \Z$ be integers.

Then $p$ divides $a^n$ iff $p^n$ divides $a^n$.

Sufficient Condition
Let $p^n \mathop \backslash a^n$.

We have $p \mathop \backslash p^n$ as $p^n = p \left({p^{n-1}}\right)$.

From the fact that Divides is Transitive, we have that $p \mathop \backslash a^n$.

Necessary Condition
Let $p \mathop \backslash a^n$.

Using Euclid's Lemma for Prime Divisors with $a_1 = a_2 = \cdots = a_n = a$ we have that $p \mathop \backslash a^n \implies p \mathop \backslash a$.

Hence $a = p r$ for some $r \in \Z$.

Raising both sides of this equation to the power $n$ we get that $a^n = p^n r^n$.

So $p^n \mathop \backslash a^n$.