3. Insert a digit (including 0) into a two-digit number to form a three-digit number (for example, inserting the digit 5 into 39 results in 359). Some two-digit numbers, when a digit is inserted, become a three-digit number that is $k$ times (where $k$ is a positive integer) the original two-digit number. What is the maximum value of $k$? ( ).
(A) 11
(B) 13
(C) 15
(D) 19
3.D.

Let the two-digit number be $10a + b$, and the three-digit number after inserting a digit $m$ be $100a + 10m + b$ (where $a$, $m$, and $b$ are integers, and $0 < a < 10$, $0 \leqslant m, b < 10$). Then,
$$
k = \frac{100a + 10m + b}{10a + b} = 10 + \frac{10m - 9b}{10a + b}.
$$

Clearly, when $a$ and $b$ are as small as possible, and $m$ is as large as possible, the value of $k$ is maximized. Therefore, when $a = 1$, $b = 0$, and $m = 9$, $k$ reaches its maximum value of 19.