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Reciprocal Symmetry

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.

Problem ID: 241 (16 Oct 2005)     Difficulty: 3 Star

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