IR-TST 2022 Problem 2

For a positive integer $n$ , let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$ , respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$ . Prove that each prime factor of $\tau(a)$ divides $\tau(b)$ .

Proposed by MohammadAmin Sharifi