Assume $A \times B$ is the smallest set that includes $M = \{(x,y) \in \mathbb{Z}^2 ; 3xy - 2x - 7y = 2 \}$. What are $A \cap B$ and $A \setminus B$?

Question

Now, I've actually solved this already: it's easy enough to see from $y = \frac{2x+2}{3x-7}$ that $|3x-7|\le |2x+2|$ because otherwise the fraction would not be an integer. This returns a set of possible $x$ that you can manually input to find all the viable $(x,y)$ pairs. Finding the two required sets from there is trivial.

However this particular problem is supposed to be solved as part of lessons in combinatorics, and that's where I'm stuck. I can see that without the term $3xy$ I could potentially use combinations with repetition, however I am unsure of how to apply that here.

Answer 1

A hint:

You can write the given equation in the form $$(3x-7)(3y-2)=20\ .$$ Now do a case analysis. Don't forget negative divisors of $20$.