When computing τ(4) as τ(2)·τ(2), why does the multiplicativity rule not apply?

Because 2 and 2 are not coprime, so the rule does not apply

Because the rule only holds for powers of 2

Because τ(2) is not an integer

Exercise · ZW-MULTIPLICATIVE-ARITHMETIC-FUNCTION-009 · ★★☆ Intermediate

When computing \(\tau(4)\) as \(\tau(2)\cdot\tau(2)\), why does the multiplicativity rule not apply?

Choose one option.

Show answer

B. Because 2 and 2 are not coprime, so the rule does not apply

Explanation

Since 2 and 2 share the factor 2, \(\gcd(2,2)=2\) and the rule requires coprimality, so \(\tau(4)=3\) and not 4.

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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.

\[f(mn)=f(m)f(n)\text{ if }\gcd(m,n)=1\]

When computing \(\tau(4)\) as \(\tau(2)\cdot\tau(2)\), why does the multiplicativity rule not apply?

Explanation

Review the concept: Multiplicative arithmetic function

More practice problems: multiplicative arithmetic function