## 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 + 12 1 + 13 1 + 14 ... 1 + 1$n$
 = 32 43 54 5 ... $n$ 1 $n$$n$ 1 $n$ + 1$n$ = $n$ + 12

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

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

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