Algorithm to find smallest number divisible by N with sum of digits as N Problem:


  Given $N$, find the smallest number divisible by $N$ whose sum of digits is equal $N$.


For example:


$n = 1$, answer is $1$
$n = 10$, answer is $190$


There is some dynamic programming algorithm, but I can't find it.
It is sequence https://oeis.org/A002998.
Assuming that "number" means "positive integer", every number divisible by $N$ must be at least $N$. Conversely, $N$ has the same digit sum as itself. Therefore the answer is simply $N$, as your two examples demonstrate.