Exercise · ZW-DIVISOR-FUNCTIONS-006 · ★★☆ Intermediate
Compute \( \tau(72) \) using the factorization \( 72=2^3\cdot 3^2 \).
Choose one option.
Show answer
- A. 12
Explanation
Since \( \tau \) is multiplicative, \( \tau(72)=(3+1)(2+1)=12 \). The option giving 9 comes from multiplying the exponents 3 and 2 instead of adding 1 to each.
Review the concept: Divisor functions
The divisor functions \( \sigma_k(n) \) and \( \tau(n) \) give, respectively, the sum of the \( k \)-th powers and the number of positive divisors of an integer \( n \geq 1 \).
\[\sigma_k(n)=\sum_{d\mid n} d^{k},\quad \tau(n)=\sigma_0(n)\]
More practice problems: divisor functions
- ZW-DIVISOR-FUNCTIONS-001 How many positive divisors does 12 have?
- ZW-DIVISOR-FUNCTIONS-002 Compute σ_1(6) , the sum of the positive divisors of 6 .
- ZW-DIVISOR-FUNCTIONS-003 State the formula for τ(p^a) for a prime p and a≥ 0 .
- ZW-DIVISOR-FUNCTIONS-004 What is σ_0(n) for any n≥ 1 ?
- ZW-DIVISOR-FUNCTIONS-005 Compute τ(7) , where 7 is prime.
- ZW-DIVISOR-FUNCTIONS-007 Compute σ_1(15) using that σ_1 is multiplicative.
- ZW-DIVISOR-FUNCTIONS-008 Compute σ_2(4) , the sum of the squares of the positive divisors of 4 .
- ZW-DIVISOR-FUNCTIONS-009 Let n=2^4· 3^2 . Compute τ(n^2) and compare it with τ(n)^2 . Are they equal?