Irreducibles in the ring of gaussian integers

Question

Let $\pi$ be an irreducible in the ring of gaussian integers that divides an odd prime. Prove that $\pi$ cannot divide both $x+2i,x-2i$ for an integer $x$.

I tried going about this by deriving a contradiction, and I think if $\pi$ divides either, then $N(\pi) = y^2 + 4$. But, I'm not sure how this helps.

Answer 1

If $\Pi$ divides $x+2i$ and $x-2i$ then it divides the difference $4i$. Since $i$ is a unit, $\Pi$ divides 2 so it cannot divide an odd prime.