Prove this discrete mathematics problem true or false

Question

Prove or disprove the following:

$$n\in \mathbb Z \; \text {is odd} \iff 8\; |\; (n^2 - 1)$$

I believe there are $2$ things to prove, but I am really lost. Any help would be appreciated thank you!

Answer 1

Suppose $8\mid (n^2-1)$; in particular $n^2-1$ is even, so $n^2$ is odd.

Suppose $n=2k+1$ is odd; then $$ n^2-1=4k^2+4k+1-1=4k(k+1) $$ Can you tell why $4k(k+1)$ is divisible by $8$?