On the number of solutions of a restricted linear congruence

Abstract

Consider the linear congruence equationa1sx1+…+aksxk≡b(mod ns) where ai,b∈Z,s∈N. Denote by (a,b)s the largest ls∈N which divides a and b simultaneously. Given ti|n, we seek solutions 〈x1,…,xk〉∈Zk for this linear congruence with the restrictions (xi,ns)s=tis. Bibak et al. [2] considered the above linear congruence with s=1 and gave a formula for the number of solutions in terms of the Ramanujan sums. In this paper, we derive a formula for the number of solutions of the above congruence for arbitrary s∈N which involves the generalized Ramanujan sums defined by E. Cohen [5].