A positive integer $N$ is appended to the right of any positive integer (for example, appending 21 to the right of 35 yields 3521). If the new number is always divisible by $N$, then $N$ is called a “magic number”. How many “magic numbers” are there among the positive integers less than 600?
Number Theory, Divisors and Multiples.
(1) If $N$ is a one-digit number;

Then by the problem, we get $N \mid 10 m + N$, so $N \mid 10 m$;
Since $m$ is any positive integer; thus $N \mid 10$;
$10 = 2 \times 5$ has $(1+1) \times (1+1) = 4$ divisors: $1, 2, 5, 10$;
Therefore, $N = 1, 2, 5$.
(2) If $N$ is a two-digit number;

Then by the problem, we get $N \mid 100 m + N$, so $N \mid 100 m$;
Since $m$ is any positive integer; thus $N \mid 100$;
$100 = 2^2 \times 5^2$ has $(2+1) \times (2+1) = 9$ divisors: $1, 2, 4, 5, 10, 20, 25, 50, 100$;
Therefore, $N = 10, 20, 25, 50$.
(3) If $N$ is a three-digit number;

Then by the problem, we get $N \mid 1000 m + N$, so $N \mid 1000 m$;
Since $m$ is any positive integer; thus $N \mid 1000$;
$1000 = 2^3 \times 5^3$ has $(3+1) \times (3+1) = 16$ divisors:
$1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000$; Therefore, $N = 100, 125, 200, 250, 500$.
In summary, there are 12 "magic numbers" among the positive integers less than 600.