I finished my number theory course in June, but there was always one question (luckily that didn't appear on my exam) that I never knew how to even attempt. It was:
Suppose $\alpha$ is a non-zero Gaussian integer. Prove that there are only finitely many congruence classes modulo $\alpha$. Obtain an upper bound for the number of such classes in terms of $\alpha$.
I was just wondering what would have been the root to take with this question because even though I've finished the course, I'd like to actually see how to do it.
Hint: Consider the square with corners $0,\alpha,i\alpha,(1+i)\alpha$. Show the every equivalence class has a representative in this square (including the border.)
Or, if you know the division algorithm in $\mathbb Z[i]$, you can use that.