Question 465281
*[tex \LARGE \frac{1}{a} + \frac{a}{b} + \frac{1}{ab} = 1]


The left side can be condensed into one fraction


*[tex \LARGE \frac{b+a^2+1}{ab} = 1]


*[tex \LARGE a^2+b+1 = ab]


We can solve for b in terms of a:


*[tex \LARGE a^2 + 1 = ab - b = b(a-1)]


*[tex \LARGE b = \frac{a^2 + 1}{a-1}]


This can be simplified by splitting up the fraction:


*[tex \LARGE b = \frac{a^2 - 1}{a-1} + \frac{2}{a-1} = (a+1) + \frac{2}{a-1}]


Here, we want the quantity 2/(a-1) to be an integer, where a is a positive integer. The only values that work are a = 2, a = 3, and solving for b:


a = 2 --> b = 5
a = 3 --> b = 5 


It can be checked that both of these solutions work.