Exercise · ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-006 · ★★☆ Intermediate
If \(\tau(n)=12\) and \(n=2^{2}\cdot 3^{?}\), what is the exponent of 3?
Choose one option.
Show answer
- A. 3
Explanation
Since \(\tau(n)=(2+1)(b+1)=12\), we get \(b+1=4\) and so \(b=3\). The option of 1 does not satisfy the divisor equation.
Review the concept: Multiplicative arithmetic function
An arithmetic function \(f\) is multiplicative if \(f(1)=1\) and \(f(mn)=f(m)f(n)\) whenever \(\gcd(m,n)=1\). Its behavior is determined by its values on prime powers.
\[f(mn)=f(m)f(n)\text{ if }\gcd(m,n)=1\]
More practice problems: multiplicative arithmetic function
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-001 What is the value of τ(12)?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-002 If f is multiplicative and gcd(m,n)=1, which expression equals f(mn)?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-003 What is the value of τ(p^3) for a prime p?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-004 Which of these functions is not multiplicative?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-005 If f is multiplicative, what is f(1)?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-007 Compute τ(2^4· 5^2).
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-008 If f is multiplicative with f(2)=3 and f(5)=4, what is f(10)?
- ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-009 When computing τ(4) as τ(2)·τ(2), why does the multiplicativity rule not apply?
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