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Math Help - Solving solutions for a set

  1. #1
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    Apr 2009
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    Solving solutions for a set

    Okay, so this is a problem that I have absolutely no idea how to start. Other than making random conjectures or saying "These are all integers!" I have no idea how to go about solving this. A pointer on where to start would be greatly appreciated.


    We're given the hint to prove that each element of the above set solves n = ax +by but I don't know how to go about doing that!
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  2. #2
    Senior Member
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    Apr 2009
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    Atlanta, GA
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    409

    Rephrase

    It may perhaps be beneficial to rephrase the question to be less technical. Let A and B be two distinct nonzero integers with GCD of G. Let N be a multiple of G. By definition of GCD, there exist integers x,y satisfying the equation Ax+By=G, therefore there exist integers x,y satisfying the equation Ax+By=N. Let x_0,y_0 be a solution to this equation. Define A'=A/G and B'=B/G. Show that:

    (i) x_0+tA' , y_0-tB' solves the above equation for all integers t

    (ii) Prove that all solutions are of this form.

    All this technical jargon makes the problem difficult to start, but see if this inspires you.
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