Let π={as(modΒ ns)}ks=0 be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results related to the covering multiplicity of π. In particular, we show that if every integer lies in more than m0=ββ ks=11/nsβ members of π, then for any a=0,1,2,β¦ there are at least subsets I of {1,β¦,k} with β sβI1/ns=a/n0. We also characterize when any integer lies in at most m members of π, where m is a fixed positive integer.