Several Results on Systems of Residue Classes

Let (m,n) and a(n) denote the g.c.d, of m, n and the residue class {x∈Z∶x≡α (mod n)} respectively. Any period of the characteristic function ofkU a_i(n_i) is called a covering period of {a_i(n_i)}_(i-1)~k.i-ITheorem Let A = {a_i(n_i)}_(i-1)~k. be a disjoint system (i. e. a_I(n_I,...,a_k(n_k) are pairwise disjoint). Let n_I,...,n_k] (the I.c.m. of n_1,...,n_k) have the prime faetorization n_1,...,n_k] = Πp_i~ai and T = Πp_iβi(β_i≥0 be the smallest positive covering period of A. Then