Which of these values can NEVER equal φ(n) for any n≥ 3?

Exercise · ZW-EULER-S-TOTIENT-FUNCTION-003 · ★★☆ Intermediate

Which of these values can NEVER equal \(\varphi(n)\) for any \(n\ge 3\)?

Choose one option.

Show answer

For \(n\ge 3\), if \(k\) is coprime to \(n\), so is \(n-k\), and \(k\neq n-k\); coprimes come in pairs, so \(\varphi(n)\) is even.

Euler's totient function \(\varphi\) assigns to each positive integer \(n\) the number of integers between 1 and \(n\) that are coprime to \(n\).

\[\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\]