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When is the value of 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.

*This problem is taken from the UKMT Mathematical Challenges.*