Discrete Mathematics · Recurrences, generating functions, and discrete asymptotics
Let n be a positive integer
Problem
Let \(n\) be a positive integer. Prove that \(\sum_{d\mid n}\varphi(d)=n\), where the sum runs over the positive divisors of \(n\).
Hint
Group the integers \(1,\dots,n\) according to their gcd with \(n\); the size of the block with gcd \(d\) is a totient.
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