Number Theory

Arithmetic Functions

Challenge Quizzes

Arithmetic Functions: Level 5 Challenges


If \( f: \mathbb{N}\mapsto \mathbb{N} \) is a bijective function that satisfies

\[ f(xy ) = f(x) f(y) \]

and \( f(2015) = 42 \), what is the minimum value of \( f(2000) \)?

Find \[\large \sum_{d|2016} \mu(d)\] where \(\mu\) denotes the Möbius function and the \(d\) are the positive divisors of \(2016.\)


If \(f(d)\) is an arithmetic function such that the equation above holds for all positive integers \(n\), find \(f(2015)\).

Notation: \(\mu\) denotes the Möbius function.

\[\sum_{\mu(n)=1} \frac{1}{n^2} = \frac{A}{B\pi^C}\]

Let \(\mu(n)\) denote the möbius function, the sum is taken over all positive integers \(n\) such that \(\mu(n)=1\), with coprime positive integers \(A\) and \(B.\) Find \(A+B+C\).


\[ \large \sum_{d|2015!}\mu(d)\phi(d).\]



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