The odd residue classes modulo 8 are . Now you just need to check that .
(Because every odd integer can be written as or .)
This is good; it has the advantage of not relying on (general) principles of modular arithmetic. However, you would have a bit more trouble generalizing this type of argument; for instance proving that the square of every integer relatively prime to 12 is congruent to 1 modulo 12 might make the argument quite a lot less elegant.
I agree this method lacks generality. But when questions are posed in such a way that the numbers "work out nice" I assume that is how it should be done.
Also, this question lends itself to induction.
Problem: Prove that if is an odd natural number then
Proof: We do this by weak induction.
Base case- Trivially
Inductive hypothesis- Assume that
Inductive step- Using the hypothesis we see that , but the next odd integer is given by and . Since is odd we know that is even therefore . So we may conclude that .