Find a positive integer satisfying the following three clues:

- When multiplied by 2 , it yields a perfect square.
- When multiplied by 3 , it yields a perfect cube.
- When multiplied by 5 , it yields a perfect fifth power.
It is reasonable to try a number of the form $2^{a} 3^{b} 5^{c}$, where $a, b$, and $c$ are nonnegative integers. We want

$$
\begin{array}{ll}
2^{a+1} 3^{b} 5^{c} \text { is a square, i.e. } & 2|a+1,2| b, \quad \text { and } 2 \mid c \\
2^{a} 3^{b+1} 5^{c} \text { is a cube, i.e. } & 3|a, \quad 3| b+1, \text { and } 3 \mid c \\
2^{a} 3^{b} 5^{c+1} \text { is a fifth power, i.e. } 5|a, \quad 5| b, \quad \text { and } 5 \mid c+1 .
\end{array}
$$

We can now search for $a, b$, and $c$ separately. We find that $n_{0}=2^{15} 3^{20} 5^{24}$ is a solution.