Color every number of form $an$ for $a=1,...,m$ with red color, and every number of form $bm$ for $b=1,...,n$ with blue color. Then all numbers from $1$ to $mn-1$ lies between some colored numbers and there are in total $m+n-1$ such groups. So at least $2$ numbers $x,y$ are in same group, but it means, that for these $x,y$ is true that $\lfloor \frac{x}{n} \rfloor = \lfloor \frac{y}{n} \rfloor$ and $\lfloor \frac{x}{m} \rfloor = \lfloor \frac{y}{m} \rfloor$