MathNet 2022-part-06 — 196 problems (EN–VI) Question 106

For any positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\frac{\tau\left(n^{2}\right)}{\tau(n)}=3$, compute $\frac{\tau\left(n^{7}\right)}{\tau(n)}$.