smc

We know that $n^2=k^3$ and $n^3=m^2$. Cubing and squaring the equalities respectively gives $n^6=k^9=m^4$. Let $a=n^6$. Now we know $a$ must be a perfect $36$-th power because $lcm(9,4)=36$, which means that $n$ must be a perfect $6$-th power. The smallest number whose sixth power is a multiple of $20$ is $10$, because the only prime factors of $20$ are $2$ and $5$, and $10=2\times 5$. Therefore our is equal to number $10^6=1000000$, with $7$ digits $\Rightarrow \boxed{7}$.