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Thread: Show that...

  1. #1
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    Show that...

    Stuck on the following guys:

    1) Show that for any positive integers a,b,c,d satisfying ab-cd=1, no cancellation is possible in the fraction a/d.

    2) Prove furthermore that no cancellation is possible in a+c/b+d.


    If i can figure out the first part, then the second, i'm guessing, will be proven in the same way...so, no need to go through with that...but i'd just like to know where to begin with this.

    My progress so far is: if i show that a and d have a gcd of 1, i.e. are relatively prime (am i using that term correctly here?), then a/d cannot be broken down. But...how to actually go about showing that....no idea really, at least not without actual values.

    Any help greatly appreciated.
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  2. #2
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    Quote Originally Posted by scorpio1 View Post
    Stuck on the following guys:

    1) Show that for any positive integers a,b,c,d satisfying ab-cd=1, no cancellation is possible in the fraction a/d.

    2) Prove furthermore that no cancellation is possible in a+c/b+d.
    Two positive number n,m are relatively prime if there exists x,y \in \mathbb{Z} such that nx+my = 1.

    (i) If ab - cd = 1 then if we pick x=b,y=-c then ax+dy=1.

    (ii) Use the same approach above. Note, b(a+c) - c(b+d) = ab + bc - bc - cd = ad - cd = 1, thus \gcd(a+c,b+d)=1.
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