What's the remainder when the sum
$1^3+2^3+3^3+\cdots+99^3$ is divided by $3$?
I saw this question on MSE but it was closed and I wanted to learn how to approach it. The help given to the asker had failed to bring the question up to an acceptable standard.
I can solve it but I doubt my methods are efficient.
One way is to start by removing every third term. Then the first two terms of the remaining sequence can be removed, and so on... a pattern may emerge.
Another way is to look at the expansion of $(x+1)^3$ and see what it does to the residues for each $x$, then sum over that by induction.
But I'm sure my inventions aren't very efficient.