[Math] Proof: $\;n^2\;$ is even if and only if $\;n\;$ is even.

Question

Please help how would you go about doing this? I'm studying for a final. This is on a study guide. I'm having a lot of trouble with this class.

Prove that $n^2$ is even if and only if $n$ is even.

Answer 1

• The forward implication: is pretty easy: take $n=2k $ so $n^2=2\times 2k^2$ is even.
• The backward implication: we have $n^2-n=n(n-1)=\text{even}\times\text{odd}=2\times\alpha$ is even so $n=n^2-2\alpha$ is even.