Solution:

Let $n$ be such a number. If $n$ is divisible by $2$ and $3$ exactly $e_{2}$ and $e_{3}$ times, then $e_{2}$ is odd and a multiple of three, and $e_{3}$ is even and one more than a multiple of three. The smallest possible exponents are $n_{2}=3$ and $n_{3}=4$. The answer is then $2^{3} \cdot 3^{4}=648$.