smc

We may let $n=7^k\cdot m$, where $m$ is not divisible by 7. Using the fact that the number of divisors function $d(n)$ is multiplicative, we have $d(n)=d(7^k)d(m)=(k+1)d(m)=60$. Also, $d(7n)=d(7^{k+1})d(m)=(k+2)d(m)=80$. These numbers are in the ratio 3:4, so $\frac{k+1}{k+2}=\frac{3}{4}⟹k=2\Rightarrow \boxed{2}$.