On solving a restricted linear congruence using generalized Ramanujan sums

Consider the linear congruence equation $x_1+\ldots+x_k \equiv b,(\text{mod } n)$for$b,n\in$\mathbb{Z}$$\mathrm{.}By$(a,b)s$,wemeanthelargest$l^s\in$\mathbb{N}$$whichdivides$a$and$b$simultaneously\mathrm{.}Foreach$d_j|n$, define ${\mathcal{C}}_{j,s}=\{1\le x\le n^s\mathrm{\mid }(x,n^s{)}_s=d_j^s\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\sum _{d\mathrm{\mid }n}{c}_{d,s}(b)\prod _{j=1}^{\tau (n)}{(c_{\frac{n}{d_j},s}(\frac{n^s}{d^s}))}^{g_j}$$ where $g_j=\mathrm{\mid }\{x_1,\dots ,x_k\}\cap {\mathcal{C}}_{j,s}\mathrm{\mid }$ for $j=1,\ldots \tau(n)$and$c{d,s}$ denote the generalized ramanujan sum defined by E. Cohen.