How to find the solutions of this equation with two variables?

Question

How could I solve the equation $5a+6b+56=ab$ ? How can I find each pairs $(a, b)$ without trying out them all, when a and b arent allowed to be negative or floating point ?

Answer 1

This is solution if you want Positive Integral solutions :

For this, you can proceed to factorize in the following way ;

$5a+6b-ab+56=0$

$(a-6)(b-5)=86$

See this link for factorization trick.

$(a-6)(b-5)=2\times 43=1\times86$

So possibilities are :

$(a-6)=2, (b-5)=43$

$(a-6)=43, (b-5)=2$

$(a-6)=1, (b-5)=86$

$(a-6)=86, (b-5)=1$

Hope this helps!