Prime Divides Power

Theorem
Let $$p$$ be a prime number.

Let $$a, b$$ be integers.

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

Proof

 * Let $$p^n \backslash a^n$$.

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

From the fact that divides is transitive, we have that $$p \backslash a^n$$.


 * Let $$p \backslash a^n$$.

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

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

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

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