here's my take on it.
we know that xy = rn = sm.
so m = rn/s = xy/s. what we'd like to do is show n/s is an integer, and then we're done.
so factor s into powers of distinct primes.
for each prime factor p of s, we have p|rn, but p does not divide r (because gcd(r,s) = 1).
thus p | n.
now if p^k | s, we must likewise have p^k | n (why?), so....