## Integer Fraction Product

#### Problem

Prove that $n$ must be odd for (1 + ^{1}/_{2})(1 + ^{1}/_{3})(1 + ^{1}/_{4})...(1 + ^{1}/_{$n$}) to be integer?

#### Solution

Begin by using the result:

$n$ + 1 $n$ | = 1 + | 1 $n$ |

So that we can write the product in the following way and cancel denominators,

1 + | 1 2 | 1 + | 1 3 | 1 + | 1 4 | ... | 1 + | 1 $n$ |

= | 3 2 | 4 3 | 5 4 | 5 | ... | $n$ 1 | $n$ $n$ 1 | $n$ + 1 $n$ | = | $n$ + 1 2 |

Hence it will be integer when $n$ is odd.

Problem ID: 114 (Apr 2003) Difficulty: 2 Star