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 be a solution to this equation. Define A'=A/G and B'=B/G. Show that:
(i) 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.