Prove a property of divisor function Let $n$ be a positive natural number whose prime factorization is $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_i$ are natural distinct prime numbers, and $a_i$ are positive nat...--prophetes.ai

Prove a property of divisor function Let $n$ be a positive natural number whose prime factorization is $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_i$ are natural distinct prime numbers, and $a_i$ are positive natural numbers. Using induction to show that the number of divisors of $n$ is $$(a_1+1)(a_2+1)\cdots(a_k+1)$$ * * *

Hint: Instead of inducting on $n$ consider inducting on $k$.