Calculate the number of divisors of an integer based on its prime factorization.
Let $n=p_{1}^{\alpha_{1}} \ldots p_{k}^{\alpha_{k}}$: choosing a divisor of $n$ amounts to choosing natural numbers $\beta_{1}, \ldots, \beta_{k}$ with $\beta_{i} \leqslant \alpha_{i}$ for all $i$. There are $\alpha_{1}+1$ choices for $\beta_{1}$ etc, so the number of divisors of $n$ is $\left(\alpha_{1}+1\right) \ldots\left(\alpha_{k}+1\right)$.