$9 \cdot 18$ is used to form a six-digit number $\overline{a b c d e f}$ with the digits $1,2,3,4,5,6$, where different letters represent different digits from 1 to 6. It is required that the two-digit number $\overline{a b}$ is a multiple of 2; and $\overline{a b c}$ is a multiple of 3; $\overline{a b c d}$ is a multiple of 4; $\overline{a b c d e}$ is a multiple of 5; $\overline{a b c d e f}$ is a multiple of 6. Find all such six-digit numbers and explain your reasoning process.
(China Beijing Junior High School Grade 2 Mathematics Competition, 1982)
[Solution] Since it is required that $\overline{a b}$ is a multiple of 2, $\overline{a b c d}$ is a multiple of 4, and $\overline{a b c d e f}$ is a multiple of 6, we have
$$
\begin{array}{l}
b, d, f \in\{2,4,6\}, \\
a, c, e \in\{1,3,5\} .
\end{array}
$$

Since $\overline{a b c d e}$ is a multiple of 5, we have $e=5$.
At this point, the 1st digit $a$ and the 3rd digit $c$ can only take 1 or 3, so $a+c=4$.
Since $\overline{a b c}$ is a multiple of 3, we have
$$
\begin{array}{c}
3 \mid a+b+c=4+b . \\
b=2 .
\end{array}
$$

Thus, we can only have
At this point, the desired six-digit number can only be found among the following numbers:
$$
\begin{array}{ll}
123456, & 123654, \\
321456, & 321654 .
\end{array}
$$

Upon inspection, since $4 \nmid 1234,4 \nmid 3214$.
Therefore, only 123654 and 321654 are the six-digit numbers that meet the requirements of the problem.