A fascinating problem . . . I haven't solved it yet.
But I have a start . . . maybe someone can follow through?
Show that there is a multiple of 2004 which has the form 22...200...0,
that is, a sequence of 2's, followed by a sequence of 0's.
It seems that a number of the form is a multiple of 501.
And all we have to do is find it . . . or prove that it exists.