how can I find Integer solutions for the two variables equation without searching factors?

Question

If $(30X+7)(30Y+1) = 50437$, then what are the integer solutions? Any way to solve it without searching $50437$ factors?

Answer 1

If you are asking for a computationally easier way, then I doubt that's possible. This is the foundation of modern cryptography. Your question is equivalent to asking "Is there an easier way of finding integer factors". Only difference being the restriction on the form of factors - $30X+7$ and $30Y+1$ Please read:

http://en.wikipedia.org/wiki/Integer_factorization

However, if you just want to avoid the technicality of factorization, you can resolve the equation in terms of $Y$, plot the function and check for values at integer points and mark the integer solution.

$$Y = \frac{\frac{50437}{30X+7} - 1}{30} $$

Anything you do in the end is equivalent to factorization.

Answer 2

If you're looking for solutions where $X$ and $Y$ are integers, we see that $(30X + 7)$ and $(30Y + 1)$ must also be integers. Hence, you need only examine the factors of $50437$.

Answer 3

Your problem is to solve $xy=50437$, i.e. factor $50437$, into factors of special form. There are clever factorization methods, that are more efficient than just "searching for factors", but one way or another you are doing the factoring.