19(1260). Find the smallest natural number that, when multiplied by 2, becomes a square, and when multiplied by 3, becomes a cube of a natural number.
Solution. Let $x$ be the smallest natural number such that $2 x=b^{2}, 3 x=c^{3}$, where $b$ and $c$ are natural numbers. From the equation $2 x=b^{2}$, it follows that $x$ is divisible by 2. Since $3 x=c^{3}$, $x$ is divisible by $2^{3}=8$ and by $3^{2}=9$, i.e., $x=2^{3} \cdot 3^{2} a^{6}=72 a^{6}$, where $a$ is any natural number. The smallest $x$ is obtained when $a=1$. Answer: 72.