5. "Lucky numbers" refer to positive integers that are equal to 19 times the sum of their digits (in decimal). Find all the lucky numbers.


5. Let $10a + b$ be a number with at most two digits. The equation $10a + b = 19(a + b)$ holds if and only if $a = b = 0$. Therefore, all lucky numbers must have at least three digits.

Assume a lucky number is an $m (m \geqslant 4)$-digit number, then the number is at least $10^{m-1}$, and the sum of its digits is at most $9m$. Therefore,
$171m \geqslant 10^{m-1}$.
When $m = 4$, $684 \geqslant 1000$ does not hold.
And $m \geqslant 5$ holds even less.
Thus, all lucky numbers are three-digit numbers.
From $100a + 10b + c = 19a + 19b + 19c$, we know
$9a = b + 2c$.
When $a = 1$, we get
$(b, c) = (1, 4), (3, 3), (5, 2), (7, 1), (9, 0)$;
When $a = 2$, we get
$(b, c) = (0, 9), (2, 8), (4, 7), (6, 6), (8, 5)$;
When $a = 3$, we get
$(b, c) = (9, 9)$;
When $a > 3$, there are no solutions.
Therefore, there are 11 lucky numbers, which are:
$114, 133, 152, 171, 190, 209, 228, 247, 266, 285$,
399.