mathschallenge.net

Problem

Given that a and b are positive integers, for which values is the following expression integer?

a/b + b/a

Solution

Let c = a/b + b/a.

Suppose that HCF(a,b) = k, such that a = kd and b = ke.

Therefore, c = d/e + e/d and HCF(d,e) = 1.

Multiplying through by d gives cd = d²/e + e. Clearly cd and e are integer, so d²/e must be integer. But as HCF(d,e) = 1, we deduce that e = 1; and by symmetry d = 1.

Hence c = d/e + e/d = 1 + 1 = 2, and as a/b = d/e = 1, we further deduce that a = b can take on any non-zero integer value.