TITLE: How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$
QUESTION [12 upvotes]: how to prove: 
$$ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$$ 
$\prod_{d|n} d$ is product of all of distinct positive divisor of $n$,
$\tau (n)$ is number (count)of all of positive divisor of $n$

REPLY [3 votes]: suppose $\tau(n)=r (I)$,$d_1,...,d_r$ is all of distinc positive divisor $n$
$d_i|n$ thus $\frac n{d_i}|n$ so {$\frac n{d_1},...,\frac n{d_r}$}$\subset${$d_1,...,d_r$}
also we have: $d_i=\frac n{\frac n{d_i}}=\frac n{d_j}$ so 
{$\frac n{d_1},...,\frac n{d_r}$}$=${$d_1,...,d_r$}
$$\prod_{d|n} d =d_1...d_r=\frac n{d_1}.....\frac n{d_r}$$ so $n^r=(d_1...d_r)^2$
$n^{\frac r2}=\prod_{d|n} d =d_1...d_r=n^{\frac {\tau(n)}2}$