Exercise · ZW-DIVISOR-FUNCTIONS-009 · ★★★ Advanced
Let \( n=2^4\cdot 3^2 \). Compute \( \tau(n^2) \) and compare it with \( \tau(n)^2 \). Are they equal?
Choose one option.
Show answer
- B. No, \( \tau(n^2)=45 \) and \( \tau(n)^2=225 \)
Explanation
We have \( \tau(n)=5\cdot 3=15 \), so \( \tau(n)^2=225 \). Since \( n^2=2^8\cdot 3^4 \), \( \tau(n^2)=9\cdot 5=45 \), so they are not equal. The option giving 15 for \( \tau(n^2) \) forgets to double the exponents.
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-006 Compute τ(72) using the factorization 72=2^3· 3^2 .
- 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 .