If $gcd(a,b)=1$ and $k \in \{{0,1,2,..ab-a-b}\}$ then there exists an integer $x$ such that $0\le x \lt b$

Question

Let $gcd(a,b)=1$ and $k \in \{{0,1,2,..ab-a-b}\}$

Since $gcd(a,b)=1$, there exist integers $x$ and $y$ such that $ax+by=k$.

Dividing both sides by $a$ and re-arranging we get $k/a=b(y/a)+x$ , then $ 0\le x \lt b$ (by the division algorithm).

My problem is that I don't know if $(k/a)$ and $(y/a)$ are integers. If I knew that $(k/a)$ was an integer then $(y/a)$ would also be, because $gcd(a,b)=1$.

Is $k/a$ an integer?

Any tips, please?