Given that :
$$\frac{1}{a} + \frac{1}{b} = \frac{1}{12} $$
With a & b integers
How can I find all possible values of a and b with only one equation (this one ?) .
From what I'v learned in math classes, I need at least 2 equation to find the values of 2 variables .
Multiplying through by $12ab$ we get $(a-12)(b-12)=144$.
There are 8 possible factorisations of 144 into two factors: $1\cdot144,2\cdot72,3\cdot48,4\cdot36,6\cdot24,8\cdot18,9\cdot16,12\cdot12$. Those give the solutions as $(a,b)=(13,156),(14,84),(15,60),(16,48),(18,36),(20,30),(21,28),(24,24)$. Obviously the first seven of those can be reversed to give $(a,b)=(156,13)$ etc.