National Math Olympiad in Slovenia

There exists a positive integer $k$ such that $5n + m = k(5m + n)$, or $(5 - k)n = (5k - 1)m$. The right-hand side is strictly positive, so the left-hand side must be strictly positive as well, which implies $5 - k > 0$ and so $k \in \{1, 2, 3, 4\}$. If $k = 1$, then $4n = 4m$, so $n = m$. If $k = 2$, then $3n = 9m$, so $n = 3m$. If $k = 3$, then $2n = 14m$, so $n = 7m$. If $k = 4$, then $n = 19m$. In each case $m$ divides $n$.