I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+br$, if I am restricted to $a,b \in \mathbb{N}$?
To be more specific, I want to show that $rs-s+v$ (where $1\le v<s<r$) can be written in this form, where $a,b\in \mathbb{N}$.
Thank you.
This is nearly the content of the Coin Problem for two denominations, which just asks which is the largest integer which cannot be realized this way, and it turns that the answer (the "Frobenius Number" for $(r, s)$) is $rs - r - s$. This result reduces your problem to checking a finite number of cases, and in practice this is easy to do by hand for small $r$ and $s$.
Edit As an answer for the question added in an edit, for each target number $m := rs - s + v$ notice that the integers $m, m - s, \ldots, m - (r - 1)s$ are inequivalent modulo $s$, precisely because $\gcd(r, s) = 1$.
Remark Maybe surprisingly, no closed-form formula exists for the Frobenius number for three or more denominations, though it's completely procedural if tedious to work it out by hand.