Proof Verification on Postage Stamp Problem (Dummit and Foote 8.1 #6)

Question

Let $a$ and $b$ be two relatively prime positive integers. Prove that the integer $ab - a - b$ cannot be written as a positive linear combination of $a$ and $b$ but that every integer greater than $ab - a - b$ is a positive linear combination of $a$ and $b$.

Let $N_0 = ab-a-b$. I want to show that for $N > N_0$, there exist $x,y$ non-negative integers for which $N=ax+by$. I already proved that $N \neq N_0$.

Since $N > N_0$, there is $k \in \mathbb{Z^+}$ for which $N = N_0 + k$. Since $(a,b)=1$, we have $as+bt=1$ for some $s,t \in \mathbb{Z}$, where without loss of generality, $s>0$ and $t<0$. Then $k=k(1)=k(as+bt) = a(ks)+b(kt)$. There are two cases.

$\textbf{Case 1:}$ $-a < kt \leq 0$. Here, we immediately have that $kt+a - 1 \geq 0$, so we can write $N = a(ks-1) + b(kt+a-1)$, where both $ks-1 \geq 0$ and $kt+a-1 \geq 0$.

$\textbf{Case 2:}$ $kt \leq -a < 0$. By the Division Algorithm, I already showed that there exist $q,r \in \mathbb{Z}$ for which $kt = aq+r$, with $-a<r\leq 0$. My question is: Where do I go from here? I am having trouble identifying $x$ and $y$ from this case.

I am having trouble finishing this last case of the problem by figuring out how to rearrange $N=a(ks-1)+b(kt+a-1)$ when $kt \leq -a < 0$. I cannot use modular arithmetic like in this question: Largest integer that can't be represented as a non-negative linear combination of $m, n = mn - m - n$? Why?