2010 AMC 12B Problem 9

To be smallest, $n$ uses only the primes of $20,$ so $n=2^a\cdot5^b$ with $a\ge2$ and $b\ge1.$ Since $n^2=2^{2a}5^{2b}$ is a perfect cube, $a$ and $b$ are multiples of $3.$ Since $n^3=2^{3a}5^{3b}$ is a perfect square, $a$ and $b$ are multiples of $2.$ Hence $a$ and $b$ are multiples of $6.$ The smallest choice is $a=b=6,$ so $n=2^6\cdot5^6=10^6=1{,}000{,}000,$ which has $7$ digits. Thus, the correct answer is E.