I want to solve $\sum_{d|n} \varphi(d)=n$ using Group theory.

Here, $\varphi(d)$ is Euler's totient function.

I think I should use $\Bbb Z_n$ and fundamental theorem of cyclic group.

Then I use $\varphi(d)$ as the number of generators of $\Bbb Z_d$

But I can't link this idea with the sum of all $ d|n $

Help me!

  • 1
    $\begingroup$ The order of a subgroup of the cyclic group is a divisor of $n$ and there is exactly one such group for each divisor. $\endgroup$ – Winther Apr 3 '16 at 14:56
  • 2
    $\begingroup$ Is math.stackexchange.com/questions/948620/… not doing it for you? $\endgroup$ – Pedro Tamaroff Apr 3 '16 at 14:56
  • $\begingroup$ @PedroTamaroff Thank you!! I understand it $\endgroup$ – Pearl Apr 3 '16 at 15:18

Hint: use the fact that, if $n=\prod p_i^{\alpha_i}, \mathbb {Z_n}\simeq \prod \mathbb {Z_{p_i^{\alpha_i}}}$.


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