3. Concatenate the natural number $N$

to the right of every natural number (exception: appending 2 to 35 results in 352). If the new number can always be divided by $N$, then $N$ is called a magic number. Among the natural numbers less than 130, the number of magic numbers is $\qquad$
3. 9

Solution: Let $P$ be any natural number, and $N$ be a magic number. Suppose $N$ is an $m$-digit number, and the number obtained by concatenation is denoted as $\overline{P N}$, then $\overline{P N}=P \times 10^{m}+N$.
$\because \overline{P N}$ can be divided by $N$, it follows that $P \times 10^{m}$ can be divided by $N$,
$\therefore$ The condition for $N$ to be a magic number is that $10^{m}$ can be divided by $N$.
$\because$ When $m=1$, $N=1,2,5$;
$$
\begin{array}{l}
m=2 \text {, } N=10,20,25,50 ; \\
m=3 \text { and } N<130 \text {, } N=100,125 .
\end{array}
$$
$\therefore$ There are 9 magic numbers less than 130.
Note: We can prove that a $k$-digit ($k$ is a natural number) magic number must be a divisor of $10^{k}$, and vice versa.

In fact, let $N$ be a $k$-digit magic number, and concatenate $N$ to the right of the natural number 1 to get: $1 \times 10^{k}+N$. According to the definition of a magic number, the number $1 \times 10^{k}+N$ can be divided by $N$, thus $1+10^{k}$ can also be divided by $N$, meaning that a $k$-digit magic number must be a divisor of $10^{k}$.
Thus we have:
One-digit magic numbers are $1,2,5$;
Two-digit magic numbers are $10,20,25,50$;
Three-digit magic numbers are $100,125,200,250,500$ and so on.
Magic numbers have many other properties, such as:
（1）If $N$ is a magic number, then $10^{m} \cdot N$ is also a magic number.
（2）The product of any two magic numbers with valid digits not being three is still a magic number.