The discussion revolves around the conditions under which the linear equation ax + by = c has integral solutions for integers x and y, particularly focusing on the implications of divisibility by an integer d.
The discussion is ongoing, with participants questioning the validity of certain steps taken in the manipulation of the equation. There is no explicit consensus on the correctness of the approach or the conclusion regarding the existence of integral solutions.
Participants are examining the implications of c not being divisible by d, which is central to the problem's conditions. There is uncertainty regarding the manipulation of the equation and the assumptions made about the divisibility of the variables involved.
HallsofIvy said:Where did the d in a/d come from? You can't just "stick" it into part of the equation.
Saying "d divides a" means a= dn for some integer n. Saying "d divides b" means b= dm for some integer m. Replace a and b in your equation by that and see what happens.