proof verification - Prove or disprove if $n$ is an odd integer then $n^2-1$ is divisible by 8

Hey I'm stuck on this question on how to prove it, I can prove if it's divisible by $4$ but I'm unsure how to do it for $8$.

Question

Prove or disprove the following:

If n is an odd integer then $n^2-1$ is divisible by 8 (and 16 is the next question).

Note

I know they're true and false respectively but I'm not sure how to prove it, if you could show me how to do both that would be greatly appreciated!

Working

$n\in\mathbb{Z^{odd}} \implies n=2k+1$ for some $k\in\mathbb{Z}$

$\therefore n^2-1=(2k+1)^2-1$

$= 4k^2+4k+1-1$

$=4k^2+4k$

$4(k^2+k)$

Since $4|8 \implies 4(k^2+k)|8$

$\therefore (n^2-1)|8$

QED

As you can see from the proof I can easily show that $(n^2-1)|4$ however I'm not sure what to do after it, as $4\nmid 8$ (I'm pretty sure)

Thanks!