Solve diophantine using modulus

Question

Find all pairs of positive integers $(m, n)$ that satisfy,

$mn + 3m - 8n = 59$

Using Modular arithmetic.

Okay, this is a diophantine equation, where can I begin?

Answer 1

Hint:

I would begin by factoring as follows $$(m-8)(n+3)=59-24=35 = 1\cdot 5\cdot7$$

Now can you think of the possibilities for $m-8, n+3$? Do not forget $m, n > 0$

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P.S. Modular arithmetic seems really unnecessary here, though it wouldn't be hard to contrive its use to solve.

Answer 2

We have $m=\dfrac{8n+59}{n+3}$

Let integer $d$ divides both $n+3,8n+59\implies d$ must divide $8n+59-8(n+3)=35$

Now for integer $m,n+3$ must divide $35\implies n+3$ must be $\in\{\pm1,\pm5,\pm7,\pm35\}$

But $n>0\iff n+3>3\implies n+3$ must be $\in\{5,7,35\}$

Test with each of the three possible values