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