VMO 2006 — Division A Question 3

Let $m,n$ be integers greater than $3$, and consider an $m\times n$ board of unit cells ($m$ rows and $n$ columns). A move places four marbles, one in each of four cells forming one of the configurations shown below. Can these moves make the number of marbles in every cell equal, in each case:
(1) $m=2004,n=2006$;
(2) $m=2005,n=2006$?