  For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod
n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes
such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1}
does not. We show that if n_k is a period of the covering function
w_A(x)=|{1\le s\le k: x\in a_s(n_s)}| then for any r=0,...,n_k-1 there are at
least m integers in the form $\sum_{s\in I}1/n_s-r/n_k$ with I contained in
{1,...,k-1}.
