Let's say I have set S and T being the set of all integer solutions to $ax+by=c$ and $ax+by=nc$ respectively, and set S* might be the same as set T.
S* = $\{ (n x_0 + n y_0) | (x_0, y_0) \in S\}$
How would I prove that S* $\subseteq$ T for all values of a,b,c,n $\in \mathbb{Z}$
To be honest, I don't even understand the question. I know that $ax+by=c$ is a diophantine equation, and that there exists a complete integer solution where $x = x_0+ \frac{b}{d}n$, $y = y_0+ \frac{a}{d}n$ $\forall n \in\mathbb{Z}$. Sadly that's as far as I can go before asking for help.