Find $m,n\in \mathbb{Z}$ s.t.
$$ \sum\limits_{k = 0}^{mn - 1} {\left( { - 1} \right)^{\left\lfloor {\frac{k} {m}} \right\rfloor + \left\lfloor {\frac{k} {n}} \right\rfloor } } = 0 $$ See here for motivation, I said something there that I can not understand now