Q:

Show $4x^3 + y^3 = 792,864,313,578,917,724,246$ has no solution for $x, y \in \mathbb{Z}$.

I think it involves something about looking at the last digits of the number and/or modular arithmetic but I don't remember how to do this. Help?

A:

Assume there is such a solution $(x,y)$. 
Since $4x^3$ and the right hands side are even, we conclude that $y$ is even. Then the left hand side is a multiple of $4$, but the right hand side is not - contradiction!

