Exercise · ZW-EULER-S-TOTIENT-FUNCTION-008 · ★★☆ Intermediate
For which of these \(n\) does \(\varphi(n)=\frac{n}{2}\) hold?
Choose one option.
Show answer
- B. 16
Explanation
Since \(16=2^4\), \(\varphi(16)=2^3=8\), exactly half of 16; the other three have odd prime factors, so \(\varphi(n)/n\) is not \(1/2\).
Review the concept: Euler's totient function
Euler's totient function \(\varphi\) assigns to each positive integer \(n\) the number of integers between 1 and \(n\) that are coprime to \(n\).
\[\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\]
More practice problems: euler's totient function
- ZW-EULER-S-TOTIENT-FUNCTION-001 What is φ(7)?
- ZW-EULER-S-TOTIENT-FUNCTION-002 What is φ(9)?
- ZW-EULER-S-TOTIENT-FUNCTION-003 Which of these values can NEVER equal φ(n) for any n≥ 3?
- ZW-EULER-S-TOTIENT-FUNCTION-004 How many residues x from 0 to 9 have a multiplicative inverse modulo 10?
- ZW-EULER-S-TOTIENT-FUNCTION-005 For which of these n does φ(n)=n-1 hold?
- ZW-EULER-S-TOTIENT-FUNCTION-006 What is φ(2)·φ(4)?
- ZW-EULER-S-TOTIENT-FUNCTION-007 What is the sum of φ(d) over all positive divisors d of 6?
- ZW-EULER-S-TOTIENT-FUNCTION-009 Which of these numbers does NOT satisfy φ(n)=12?
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