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Math Help - Proving that no cancellation is possible in the fraction (a+c)/(b+d), if ab-cd=1

  1. #1
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    Proving that no cancellation is possible in the fraction (a+c)/(b+d), if ab-cd=1

    Hi,

    I have the following problem, but I don't know what to do. I have been told by someone to think about gcd(a+c, b+d), but I still don't know how to prove this. Thanks for your help!

    "Prove that for any positive integers a, b, c, d satisfying

    ab-cd=1

    no cancellation is possible in the fraction

    (a+c)/(b+d), where the result is a rational number."
    Last edited by mbmstudent; February 27th 2011 at 12:01 PM. Reason: I typed "irrational," when it was supposed to be "rational"
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  2. #2
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    The set of rationals is closed under addition and nonzero division. Thus there is no way that the final expression you gave could be irrational.
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  3. #3
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    I am sorry, I got confused with another exercise so I typed it wrong... It is supposed to be rational :-/
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  4. #4
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    Perhaps start by solving the equation for a, and substituting to get \frac{1+c(b+d)}{b(b+d)}.
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    What you want to show is that a+c, b+d are relatively prime.

    It suffices to notice that b(a+c)-c(b+d)=1.
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