A friend of mine (who does not speak English) asked me if I can help him with one of his math problems, so I am going to explain it here. If there is something, that does not make sense anyhow, I will edit it.
We are dealing with diophantine equations of the following type:
$$axy + cx + by = d \tag{1}\label{1}.$$
Even though it may look like a quadratic equation with constants equal to 0, in fact, it is a hyperbolic relation, where $y=(d-bx)/(ax+c)$. The following holds: if $x$ raises, $y$ decreases hyperbolically.
Basically, the equation $\ref{1}$ is derived from the following expression:
$$ (ax + b) * (ay + c) $$
in such way, that $(ax+b) = N$ and $(ay+c)= P$. Therefore, after simple adjustments, we get:
$$a^2xy+acx+aby = N*P - bc.\tag{2}\label{2}$$
After dividing \ref{2} by $a$ and providing the following substitution $\frac{N*P - bc}{a} = d$, we get the equation \ref{1}.
We can get the classic solution for equations of this type by getting the dividers of $(ad+bc)$. If $(ad+bc) = N*P$, then some of these dividers dividers (at least 2) are solutions of the equation \ref{2}. However, that's a general solution of such problem.
I have 400-600 equations of the already mentioned type and I have the concrete values of the constats $a, b, c, d$ for all of these equations. However, only few of them have a solution and I would like to know which of them these are. Therefore, since I need to know only which of these equations are solvable, I would need to know if there exists any solvability condition of these equations in order to make my problem solving efficient.
I state one of the concrete examples of my equations, for which I need to find out, whether there exists a solution or not. As I already mentioned, I do not care, what is the solution, but I only care, whether there's a solution for such equation:
$$a = 3\,000 , b = 667 , c = 661, d = 377 608,$$
so the equation is
$$ 3\,000xy+667x+661y = 377\,608.$$
Apply SFFT:
$3000^2xy + 3000 \times 667 x + 3000 \times 661 y +667\times 661= 3000 \times 377608 + 667\times 661$
$(3000x+667)(3000y+661) = 818495$
It remains to factor the RHS and see if we can get it into the form of the LHS.
Working mod $a = 3000$ allows us to conclude that there are no solutions if we reach a contradiction. E.g. In this case, we have $2887 \equiv 2495 \pmod{3000}$, so there are no solutions.
Note that even if the modulo equation is true, it does not imply that solutions must exist.