Interesting problem. If r/s = t/u, then there must be a rational number m/n in lowest terms such that rm/n = t and sm/n = u. Question: what if n = 1? If n is not 1, then n divides both r and s because gcd(m, n) = 1.
Let r, s, t, u be integers >= 1. Suppose r/s = t/u and both fractions are in lowest terms. Prove that r = t and s = u.
This problem seems like it should be obvious, but I cannot come up with a way to start it. I'm thinking that I might make use of gcd(r,s) = gcd(t,u) = 1, or by somehow showing that r & t and s & u have the same prime factorizations, but I'm not sure where to begin.