Prove that $\prod_{d|n} d = n^{\frac{τ(n)}{2}}$
Prove that $\prod_{d|n} d = n^{\frac{τ(n)}{2}}$
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number-theory
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2You should both type using LaTeX, to make your stuff crystal clear, and add some explanations, e.g.: it "seems obvious" that $\,\tau\,$ is some kind of arithmetical function...but which one? – 2012-10-18
1 Answers
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$d$ is a divisor of $n$ if and only if $\frac{n}{d}$ is a divisor of $n$.
Then
$\prod_{d|n} d =\prod_{d |n} \frac{n}{d}$
Thus,
$ \left( \prod_{d|n} d \right)^2= \prod_{d|n} d \cdot \prod_{d |n} \frac{n}{d}= \prod_{d|n} n$
You don't need to split it in to cases, that split was probably suggested by someone who expected you to group the $d$ and $\frac{n}{d}$ terms in $\prod_{d|n} d$... In that case, if $n=k^2$, you cannot group $k$ with $\frac{n}{k}=k$....
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0Yes, it works in that case, but there is no need to discuss the two cases separately ;) – 2012-10-18