Competition problem — Number theory

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Number theory Difficulty 6.8 National olympiad Prove it

Theorem 2 For any positive integer $m$ ,
$\sum_{d \mid m} \varphi(d)=m$

We provide two proofs of the theorem. The first proof uses formula (21) from Chapter 1, §5, and the recently proven formula (4). The second proof derives the result by analyzing the relationship between complete residue systems and reduced residue systems.

Solution

None

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