The following is a Home Work question provided by our instructor:
let a,b,c be positive integers. Prove that if a+b>c then the equation ax+by = c has no solutions in positive integers.
I think the statement is false since $4x+12y=4$ is a counter example. However, it is wired that a home work question be formulated that way. Or, did I miss something?
The instructor is interested in positive integers,which are {1,2,3,4,....}
If $ a+b>c$ and $x\ge 1$ and $y\ge 1$, then $ax+by \ge a+b >c$
Thus the equality does not happen, that is there is no solution in positive integers.