Questions: Divisor Functions and Multiplicative Functions

Question 1

What is τ(360)? (Note: 360 = 2³ × 3² × 5)

• A. 10 — sum of the exponents plus one
• B. 12 — product of the exponents
• C. 24 — product of each exponent plus one
• D. 15 — number of distinct prime factors times 5

Answer: (C)

Multiplicativity gives τ(n) = (a₁ + 1)(a₂ + 1)... for n = p₁^a₁ · p₂^a₂ ·... So τ(360) = τ(2³ · 3² · 5¹) = (3+1)(2+1)(1+1) = 4 · 3 · 2 = 24. Each prime power contributes an independent factor because the divisors of n are formed by independently choosing one divisor from each prime component. Option A confusingly adds exponents instead of multiplying; option B multiplies them without adding 1.

Question 2

You know τ(m) = 6 and τ(n) = 4. Under what condition can you conclude τ(mn) = 24?

• A. Always — multiplication of divisor counts is always valid
• B. Only when m and n are both prime powers
• C. Only when gcd(m, n) = 1
• D. Only when m and n share no prime factors that appear in the key factorization

Answer: (C)

Multiplicativity states: if gcd(a, b) = 1, then τ(ab) = τ(a)τ(b). The coprimality condition is essential — without it, shared prime factors cause divisors to be counted multiple times. For example, τ(4) = 3 and τ(6) = 4, but τ(24) = 8, not 12, because gcd(4, 6) = 2 ≠ 1. Option A is the classic error — blindly multiplying divisor counts without checking coprimality.

Question 3

τ(mn) = τ(m) × τ(n) is true for most positive integers m and n.

• T. True
• F. False

Answer: False

This holds only when gcd(m, n) = 1. When m and n share common prime factors, their divisors interact and τ(mn) < τ(m)τ(n). Counterexample: τ(4) = 3 and τ(6) = 4, but τ(24) = 8, not 12. The multiplicativity property is powerful precisely because it applies across coprime factors — which is why the prime factorization (whose prime power components are always pairwise coprime) is the right framework for computing τ.

Question 4

A number n is called perfect when σ(n) = 2n — its positive divisors (including itself) sum to twice n.

• T. True
• F. False

Answer: True

This is the classical definition of a perfect number. For n = 6: divisors are 1, 2, 3, 6, so σ(6) = 12 = 2 × 6. For n = 28: σ(28) = 1+2+4+7+14+28 = 56 = 2 × 28. The connection to σ is immediate: σ(n) sums ALL positive divisors of n including n itself, so σ(n) = 2n means the sum of proper divisors (all divisors except n itself) equals n.

Question 5

Explain why multiplicativity means τ(n) is completely determined by the prime factorization of n, and give the resulting formula.

Model answer: Because prime powers in the factorization are pairwise coprime, multiplicativity applies at every step: τ(p₁^a₁ · p₂^a₂ · ...) = τ(p₁^a₁) · τ(p₂^a₂) · .... And τ(p^a) = a + 1 for any prime p, since the divisors of p^a are exactly 1, p, p², ..., p^a. So τ(n) = (a₁ + 1)(a₂ + 1)... — the entire function is determined by the exponents in the prime factorization.

Multiplicativity means a function on all integers is determined by its values on prime powers, because every integer factors into coprime prime powers. For τ, the value at each prime power p^a is easy to count directly (a+1 divisors). The formula τ(n) = (a₁+1)(a₂+1)... chains multiplicativity across all prime factors. This prime-by-prime independence is the structural pattern shared by σ, Euler's totient φ, the Möbius function μ, and all multiplicative functions in number theory.