Exercise · ZW-MOBIUS-FUNCTION-005 · ★★☆ Intermediate
How many positive divisors of 12 satisfy \(\mu(d)\neq 0\)?
Choose one option.
Show answer
- B. 4
Explanation
The divisors of 12 are 1, 2, 3, 4, 6 and 12; only 4 and 12 contain a squared prime, so 4 of them have \(\mu(d)\neq 0\).
Review the concept: Möbius function
The Möbius function \(\mu(n)\) is an arithmetic function: it equals \(1\) if \(n=1\), \((-1)^k\) if \(n\) is a product of \(k\) distinct primes, and \(0\) if \(n\) has a repeated prime factor.
\[\sum_{d\mid n}\mu(d)=\begin{cases}1 & n=1\\ 0 & n>1\end{cases}\]
More practice problems: möbius function
- ZW-MOBIUS-FUNCTION-001 What is μ(1)?
- ZW-MOBIUS-FUNCTION-002 What is μ(7)?
- ZW-MOBIUS-FUNCTION-003 Which of these numbers has μ(n)≠ 0?
- ZW-MOBIUS-FUNCTION-004 If n is a product of 4 distinct primes, what is μ(n)?
- ZW-MOBIUS-FUNCTION-006 You know that g(n)=∑_d| n f(d) for every n, with g(1)=2, g(2)=5, g(3)=4 and g(6)=9. What is f(6)?
- ZW-MOBIUS-FUNCTION-007 What is μ(10)·μ(3)?
- ZW-MOBIUS-FUNCTION-008 Which of these statements about μ is false?
- ZW-MOBIUS-FUNCTION-009 If p is prime and k≥ 2, what is μ(p^k)?