\[d | n\]
then \[d, \; \frac{n}{d}\]
are both divisors of \[n\]
and so \[\prod_{d | n} d= \prod_{d | n} \frac{n}{d}\]
. There are \[\tau (n)\]
divisors of \[n\]
so\[\prod_{d | n} d \prod_{d | n} \frac{n}{d} = \prod_{d | n} d \frac{n}{d} = \frac{d | n} n= n^{\tau (n)} \]
where
\[\tau (n)\]
is Euler's Totient Function.
Hence \[\prod_{d | n} d= n^{\frac{\tau (n)}{2}}\]
.