Producing an Integer
Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?
Age
14 to 16
Challenge level
Problem
For which values of $n$ is the product $$\left(1 + {1 \over 2}\right)\left(1 + {1 \over 3}\right)\left(1 + {1 \over 4}\right)...\left(1 + {1 \over n}\right)$$ equal to an integer?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: When $n$ is odd
The product is $${3 \over 2}\times{4 \over 3}\times{5 \over 4}\times \cdots\times{(n+1) \over n}$$ which will reduce to $${(n+1) \over 2}$$i.e. it is an integer only when n is odd.