jmc

Note firstly that $18=2\cdot 3^2$, so $n$ must be divisible by both $2$ and $3$. Furthermore, $640=2^7\cdot 5$, so $n$ must be divisible by $2^3$ and $5$, since the smallest power of 2 that, when cubed, is no smaller than $2^7$ is $2^3$. Therefore, $n$ must be divisible by $2^3$, $3$, and $5$. Note that $2^3\cdot 3\cdot 5=120$ is the smallest possible integer that satisfies all of these conditions, so we have $n=\boxed{120}$.