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About tico numbers

Source: Central American Olympiad 2003, Problem 6

June 1, 2007
number theory proposednumber theory

Problem Statement

Say a number is tico if the sum of it's digits is a multiple of 20032003.
(i)\text{(i)} Show that there exists a positive integer NN such that the first 20032003 multiples, N,2N,3N,…2003NN,2N,3N,\ldots 2003N are all tico.
(ii)\text{(ii)} Does there exist a positive integer NN such that all it's multiples are tico?
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