On the range of a covering function

Let {as(modns)}s=1k(k>1) be a finite system of residue classes with the moduli n1,…,nk distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1⩽s⩽k:x≡as(modns)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.