OMCC 2003 Problem 6

Say a number is tico if the sum of it's digits is a multiple of $2003$ . $\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico. $\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?