The Monochromatic Column problem

Let p1p1, p2p2,…, pnpn be a sequence of nn pairwise coprime positive integers, and let P=p1p2P=p1p2…pnpn. Let 0,1,…,m-10,1,…,m-1 be a sequence of m different colors. Let A   be an n×mPn×mP matrix of colors in which row i   consists of blocks of pipi consecutive entries of the same color, with colors 0 through m-1m-1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A   in which every entry is the same color. The solution for m=3m=3 colors is presented and proved.