# Math Help - Unique Irreducible Fractions

1. ## Unique Irreducible Fractions

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.

2. 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.

3. Suppose we have $\frac{r}{s}=\frac{t}{u}$, where both fractions are in lowest terms. Then, $ru=st$, which translates to $r|st$, $u|st$, $s|ru$, and $t|ru$. Since $\gcd(r,s)=\gcd(t,u)=1$, we must conclude that $r|t$, $u|s$, $s|u$, and $t|r$. This implies that $r=t$ and $s=u$.