I have tried to prove that $9^n - 4^n \equiv 0 \pmod{5}$.
At first I started out with considering the cases when $n$ is even and when it's odd and then show that the resulting expressions are congruent with $0 \pmod{5}$, but i think the proof can be shortened even further:
$$ 9^n - 4^n \equiv (-1)^n - (-1)^n $$
No matter what value $n$ takes, $a - a = 0$ for all $a$, QED.
Is this reasoning solid?