Prove or disprove: if τ(m)=τ(n) for positive integers m,n , then m=n . Give a counterexample if it is false.

It is false: τ(6)=τ(10)=4 with 6≠ 10

It is true, because the divisors determine the number

It is true, because τ is injective

Exercise · ZW-DIVISOR-FUNCTIONS-010 · ★★★ Advanced

Prove or disprove: if \( \tau(m)=\tau(n) \) for positive integers \( m,n \), then \( m=n \). Give a counterexample if it is false.

Choose one option.

Show answer

A. It is false: \( \tau(6)=\tau(10)=4 \) with \( 6\neq 10 \)

Explanation

We have \( \tau(6)=\tau(10)=4 \) with \( 6\neq 10 \), so the claim is false. The option claiming \( \tau(2)=\tau(4) \) is wrong, since \( \tau(2)=2 \) and \( \tau(4)=3 \).

Study Elementary number theory in ZWMath →

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 \( n \geq 1 \).

\[\sigma_k(n)=\sum_{d\mid n} d^{k},\quad \tau(n)=\sigma_0(n)\]

Prove or disprove: if \( \tau(m)=\tau(n) \) for positive integers \( m,n \), then \( m=n \). Give a counterexample if it is false.

Explanation

Review the concept: Divisor functions

More practice problems: divisor functions