Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, it can be shown that the intersection of these two arithmetic progressions is another arithmetic progression $N=(ac)n+c\delta+d$ where $\delta\in\mathbb{N}:a|\left(c\delta+d-b\right)$. Is there a way to transform $\delta$ such that the condition of divisibility is eliminated?

Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, where $n\in\mathbb{N}^0$, it can be shown that the intersection of these two arithmetic progressions is another arithmetic progression $N=(ac)n+c\delta+d$ where $\delta\in\mathbb{N}:a|\left(c\delta+d-b\right)$.

For example the intersection of $N=5n+3$ and $N=7n-2$ by the above formula is $N=35n+33$

Is there a way to transform $\delta$ such that the condition of divisibility is eliminated?