If $3\mid a$, then $3\mid a^2$

Question

I want to prove for all integers $a$ that if $3\mid a$ then $3\mid a^2$.

Suppose that $a \in \mathbb{Z}$ and $3\mid a$. Then $a = 3k$ for any integer k. Hence $$a^2 = (3k)(3k) = 9k^2 = 3(3k^2)$$

Since $k$ is a integer, $3k^2$ is an integer. Thus $a^2 = 3b$ for some integer $b$. Therefore $3\mid a^2$.

Is this proof correct?

Answer 1

Your proof is absolutely correct. That is the usual way to prove this. You can simply extend the this to prove

If $c|a$ then $c|a^2$.

And in a similar manner you can prove that $c^2|a^2$