In this paper we study the distribution of the infinite word t_q,n := (s_q(k) mod n)_k=0^∞, which we call the generalized Thue-Morse sequence. Here s_q(k) is the digit sum of k in base q. We give an explicit formulation of the exact minimal value of M such that every M consecutive terms in t_q,n cover the residue system of n, i.e., {0, 1, ... , n-1}. Also, we prove some stronger related results.