On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequencesthe covering function is w(x)= |{1 s k : x a_s (modn_s)}|. Using equalitiesinvolving roots of unity we characterize those systems with afixed covering function w(x). From the characterization we revealsome connections between a period n₀ ofw(x) and the modulin₁, .. . , n_k in such a systemA. Here are three centralresults: (a) For each r=0,1,. . .,n_k/(n₀,n_k)–1 there exists aJc{1, . . . ,k–1} such that . (b) Ifn₁···n_k–l <n_k–l+1 =···=n_k (0 <l <k), then for any positiveinteger r <n_k/n_k–l withr 0 (modn_k/(n₀,n_k)), the binomialcoefficient can be written as thesum of some (not necessarily distinct) prime divisors ofn_k. (c)max_(xw(x)can be written in the form wherem₁, .. .,m_k are positiveintegers.The research is supported by the Teaching andResearch Award Fund for Outstanding Young Teachers in HigherEducation Institutions of MOE, and the National Natural ScienceFoundation of P. R. China.