422. The quotient of dividing a three-digit number by the sum of its digits is 13, and the remainder is 15. Find all such three-digit numbers.
Let's denote the desired number as $\overline{x y z}$. We obtain:

$$
100 x + 10 y + z = 13(x + y + z) + 15, \quad 87 x = 3 y + 12 x + 15, \quad 29 x = y + 4 z + 5
$$

Since

$$
y + 4 z + 5 \leqslant 9 + 4 \cdot 9 + 5 = 50
$$

then $29 x \leqslant 50$, from which $x = 1$.

Now let's solve the equation

$$
29 = y + 4 z + 5, \quad y + 4 z = 24
$$

From this, we see that the digit $y$ is divisible by 4. We find the following solutions:

$$
y = 0, z = 6 ; \quad y = 4, z = 5 ; \quad y = 8, z = 4
$$

Answer: $106, 145, 184$.