KöMaL problem A, 2020

Each of $N$ people chooses a random integer number between 1 and 19 (including 1 and 19, and not necessarily with the same distribution). The random numbers chosen by the people are independent from each other, and it is true that each person chooses each of the 19 numbers with probability at most $99\%$. They add up the $N$ chosen numbers, and take the remainder of the sum divided by 19. Prove that the distribution of the result tends to the uniform distribution exponentially, i.e. there exists a number $0<c<1$ such that the mod 19 remainder of the sum of the $N$ chosen numbers equals each of the mod 19 remainders with probability between $1/19-c^N$ and $1/19+c^N$.