Sign up to access problem solutions.

Already have an account? Log in here.

It might seem counterintuitive, but if you convolute these complicated things, they'll simplify in no time! Extend your understanding of the totient function by studying this class of functions.

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 \( f(2000) \)?

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

\[\sum_{d|n}\mu\left(\frac{n}{d}\right)f(d)=n\]

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.

Sign up to access problem solutions.

Already have an account? Log in here.

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

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 \(A+B+C\).

Sign up to access problem solutions.

Already have an account? Log in here.

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

**Notations**

- \(\mu(d)\) denotes the Möbius function.
- \(\phi(d)\) denotes Euler's totient function.
- \(d\) are the positive factors of \(2015!.\)

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...