1. The problem statement, all variables and given/known data Prove: a/b + b/a equals an integer, for any a,b nonzero positive integers, if and only if a = b 2. Relevant equations Divisibility: d|e implies e = dj where j is an element of j for d and e element of the integers. 3. The attempt at a solution 1. Let a/b + b/a = k where k is an element of the integers 2. a/b + b/a = (a^2 + b^2) / ab 3. (a^2 + b^2) = kab 4. Let (a^2 + b^2) = c where c is an integer 5. c = kab 6. Step five implies the three following things: a|c , b|c , ab|c 7. If a|c and b|c, then ab does not divide c and therefore step 6 demonstrates a contradiction. 8. Therefore a must equal b Is this a solid proof? If not, could you tell me where the weakness is? EDIT : If a and b are relatively prime. I cannot use my contradiction then! Any ideas ? Thank you for your time.