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I need to prove the proposition.

Let $a$ be an arbitrary integer. Then for every positive integer $n$, we have $$\sum_{d \mid n}\phi\left(\frac{n}{d}\right)a^d\equiv0\pmod{n}.$$

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    I used the Burnside's Lemma (Cauchy-Frobenius), this is Group and rings Theory applied, but I want to know if I can use just number theory, anyone tolds me that I can find a solution to this problem using "Pólya counting" But I don't understand it2012-10-20
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    You shouldn't think of mathematical subjects as being divided like that. Burnside's lemma is Burnside's lemma. You could call it group theory. It would be equally valid to call it combinatorics. This particular application could safely be called number theory. The labels don't matter. Polya enumeration is a special case of Burnside's lemma.2012-10-20
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    I posted just now, but didn't see the two comments that appeared very recently. I hope my post is useful in spite of the overlap with these comments.2012-10-20
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    Check it directly when $n$ is a prime power. For other $n$, pick a prime factor $p$ of $n$ and write $n$ as a product of a $p$-power and another number relatively prime to $p$. Divisors of $n$ break up in a similar way to this, and $\varphi$ is multiplicative, so you should be able to derive the result by induction on the number of different prime factors of $n$.2013-01-25

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