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Number theory

Source: Serbian national olympiad 2022 P4

April 7, 2022
number theoryphi functionSerbian competitioninequalities

Problem Statement

Let f(n)f(n) be number of numbers x∈{1,2,⋯ ,n}x \in \{1,2,\cdots ,n\}, n∈Nn\in\mathbb{N}, such that gcd(x,n)gcd(x, n) is either 11 or prime. Prove ∑d∣nf(d)+φ(n)≥2n\sum_{d|n} f(d) + \varphi(n) \geq 2n For which nn does equality hold?
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