# Solving solutions for a set

• June 3rd 2009, 02:12 PM
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.

http://img29.imageshack.us/img29/2483/helpa.png
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!
• June 3rd 2009, 05:25 PM
Media_Man
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. (Happy)