Let $k$ be an odd integer. As a part of an introductory class to proofs, I wanted to show that the number $k^2 - 1$ is divisible by $8$, and managed to do this by checking that it is congruent modulo $8$. However a student came to me today asking if it was possible to prove the claim by using the greatest common divisor of the two numbers $8$ and $k^2 - 1$, but I couldn't answer them since I have very little experience in number theory and the question came right out of the bushes.
Is there a proof to this claim, that a high school student who is fairly fluent in math could grasp, and that uses the $\gcd$ to its advantage somehow? To clarify, the student tried using the fact that if $a$, $b$, $c$ and $d$ are numbers such that $a = bc + d$, then $\gcd(a,b) = \gcd(b,d)$.