3. A letter mixer in the printing press has jumbled the digits $0,2,3,4,4,7,8,8,9$ with which a number is written, which is the sixth power of some natural number. What is that number?
Solution. Let $n$ denote the desired number. The sum of the digits of the number $n^{6}$ is equal to 45, so it is divisible by 9, which means that the number $n$ is divisible by 3. Since $21^{6}$ is an eight-digit number, and $33^{6}$ is a ten-digit number, we have $n \in\{24,27,30\}$. However, $n \neq 30$, because the number $30^{6}$ has six zeros in its notation. Similarly, $n \neq 24$, because the unit digit of the number $24^{6}$ is 6, and the digit 6 does not appear in the notation of the number $n^{6}$. Finally, from $27^{6}=387402489$, we get $n=27$.