13. (6 points) Mathematician Wiener is the founder of cybernetics. At the ceremony where he was granted his Ph.D. from Harvard University, someone, curious about his youthful appearance, asked him about his age. Wiener's response was quite interesting. He said, “The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together use all the digits from 0 to 9 exactly once, without repetition or omission.” So, how old was Wiener that year? (Note: The cube of a number \(a\) is \(a \times a \times a\), and the fourth power of a number \(a\) is \(a \times a \times a \times a\))
18
【Solution】First, use estimation to roughly determine the age range of Wiener. According to $17^{4}=83521, 18^{4}=$ $104976, 19^{4}=130321$, based on the problem, we can conclude: his age is greater than or equal to 18 years old; then, $18^{3}=5832, 19^{3}=6859, 21^{3}=9261, 22^{3}=10648$, indicating that Wiener's age is less than 22 years old. According to these two ranges, it could be one of the numbers: $18, 19, 20, 21$.
Since $20, 21$ have the last digits of $0, 1$ respectively, whether it is the cube or the fourth power, it does not match the condition of "all 10 digits are used, neither repeated nor missing", so they can be ruled out.
Only 18 and 19 are left. Try them one by one,
$$
\begin{array}{c}
18 \times 18 \times 18=5832, \quad 18 \times 18 \times 18 \times 18=104976 ; \\
19 \times 19 \times 19=6859, \quad 19 \times 19 \times 19 \times 19=130321 ;
\end{array}
$$

The one that meets the requirement is 18.
Therefore, the answer is: 18.