Question 43
For a positive integer $n \gt 2$, consider the $n-1$ fractions
$$
\frac{2}{1}, \frac{3}{2}, \cdots, \frac{n}{n-1}
$$
The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal $1$? Find all values of $n$ for which this is possible and prove that you have found them all.
$$
\frac{2}{1}, \frac{3}{2}, \cdots, \frac{n}{n-1}
$$
The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal $1$? Find all values of $n$ for which this is possible and prove that you have found them all.