A formula for the number of solutions of a restricted linear congruence

Consider the linear congruence equation x1 + . . .+ xk ≡ b (mod n ) for b ∈ Z, n, s ∈ N. Let (a, b)s denote the generalized gcd of a and b which is the largest l s with l ∈ N dividing a and b simultaneously. Let d1, . . . , dτ(n) be all positive divisors of n. For each dj | n, define Cj,s(n) = {1 6 x 6 n s : (x, n)s = d s j}. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on xi. We generalize their result with generalized gcd restrictions on xi and prove that for the above linear congruence, the number of solutions is 1 ns ∑