5. [4] Find the smallest positive integer that is twice a perfect square and three times a perfect cube.
Answer: 648. 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 $2^{3} \cdot 3^{4}=648$.