Modulo Proof Help? I'm working on a proof for my number theory course and I am a bit confused on how to prove a certain case of the question..


*

*If $n$ is a odd positive integer or if $n$ is divisible by $4$ then $$1^3 +
   2^3 + 3^3 + ... + (n-1)^3 \equiv 0 \pmod n$$is this statement
true if $n$ is even but not divisible by $ 4$?


So for the case "$n$ is a positive integer divisible by $4$"
I set $n = 4k$ and try to plug $n$ into $\frac{n^{2}(n-1)^{2}}{4}$
But I can't seem find a way to simplify it and otherwise prove its congruent to $0 \pmod n$.

If $n=4k$ then $n^2 = 4kn$; hence $n^2 (n-1)^2 / 4 = n \cdot k \cdot (n-1)^2$ which is divisible by n.
If n is odd, $(n-1)^2$ is 0 mod 4, so $n^2(n-1)^2/4 = n (n(n-1)^2/4) which is 0 mod n.
