$1 \cdot 13$ If a natural number $N$ is appended to the right of every natural number (for example, appending 2 to 35 gives 352), and if the new number is always divisible by $N$, then $N$ is called a magic number. Among the natural numbers less than 130, how many magic numbers are there?
(China Junior High School Mathematics League, 1986)
［Solution］Take any natural number $P$, let $N$ be an $m$-digit magic number.
Let the number formed by writing $N$ to the right of $P$ be $\overline{P N}$, then
$$
\overline{P N}=P \cdot 10^{m}+N \text {. }
$$

By the problem statement, if $N$ is a magic number, then

Thus,
$$
\begin{array}{c}
N \backslash \overline{P N}, \\
N \mid P \cdot 10^{m} .
\end{array}
$$

Since $P$ is any natural number, it follows that
$N$ । $10^{m}$.

When $m=1$, $N=1,2,5$.
When $m=2$, $N=10,20,25,50$.
When $m=3$, and $N<130$, $N=100,125$.
Therefore, there are 9 magic numbers less than 130.