$5.1,2, \cdots, 9$ each appear once to form a nine-digit number called a "good number". It is known that the sum of nine good numbers (not necessarily distinct) is an integer multiple of $10^{k}$. Find the maximum value of $k$.

A nine-digit number formed by $1,2, \cdots, 9$ each appearing once is called a “good number”. Given that the sum of nine good numbers (not necessarily distinct) is a multiple of $10^{k}$. Find the maximum value of $k$.
5. The maximum value of $k$ is 8.

Since the sum of the digits of each good number is 45, which is a multiple of 9, the good number is a multiple of 9. Therefore, the sum $S$ of nine good numbers is also a multiple of 9.
Because $10^{k} \mid S$, it follows that $9 \times 10^{k} \mid S$.
This implies,
$$
\begin{array}{l}
9 \times 10^{k} \leqslant S \leqslant 9 \times 987654321 < 9 \times 10^{9} \\
\Rightarrow k < 9. \\
\text{Also, } 8 \times 987654321 + 198765432 = 81 \times 10^{8},
\end{array}
$$

Thus, the maximum value of $k$ is 8.