If and only if $m \equiv n(\bmod k)$, or $m \equiv -n(\bmod k)$ and $\left[ {\frac{m}{k}} \right]k \geqslant n$.

yunxiu wrote: If and only if $m \equiv n(\bmod k)$. I don't think so because I have found another satisfying tiling for $(5,13,2)$ or $(3,13,8)$ but $(7,13,8)$ doesn't satisfy. I don't know how to upload picture for my tiling, but I guarantee that is true