## 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` = `d`^{2}/`e` + `e`. Clearly `cd` and `e` are integer, so `d`^{2}/`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