$Ax + By = C$ where $A, B, C$ are positive whole numbers. Is it possible to have an infinite number of solutions to this?

Question

$Ax + By = C$ where $A, B, C$ are positive whole numbers. Is it possible to have an infinite number of solutions to this or will it have a specific number of solutions or just a single solution? (solutions being positive whole numbers too, i.e. both $x$ and $y$ should be positive whole numbers too.)

Answer 1

There can be no solution (e.g. $x+y=1$), exactly one solution (e.g. $x+y=2$), or multiple solutions (e.g. $x + y = 3$).

In any case, the number of solutions is finite, since $0 < x < \frac CA$ and $0 < y < \frac CB$.

Answer 2

Since all numbers are positive, we have $$1\le x,y <Ax+By,\quad\text{hence}\quad 1\le x,y\le C-1,$$ so there can't be more than $(C-1)^2$ solutions.

Answer 3

Since $B\neq0$ any $x$ there is at most one $y$ that satisfies $Ax+By=C$.

Also for $x$ large enough we have $Ax>C$ so that $Ax+\text{ any positive number}>C$

Since $x$ is demanded to positive we conclude that there are only a finite number of $x$'s for wich there is an $y$ that satisfies $Ax+By=C$.