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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

times...times
1 +
1

n

=
3

2
times
4

3
times
5

4
times
 

5
times...times
n minus 1

 
times
n

n minus 1
times
n + 1

n
=
n + 1

2

Hence it will be integer when n is odd.

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

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