ELMO SL 2017 C1

Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$ , and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$ . For each $\{m,n\}$ , determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$ , or prove that no such $k$ exists.

[i]Proposed by Michael Ren