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Math Help - Unique Irreducible Fractions

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
    Senior Member roninpro's Avatar
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    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.
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