| Abstrakt: |
Abstract: Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\pmod{n^s}$ for $b\in\mathbb Z$, $n,s\in\mathbb N$. Let $(a,b)_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in\mathbb N$ dividing $a$ and $b$ simultaneously. Let $d_1,\ldots, d_{\tau(n)}$ be all positive divisors of $n$. For each $d_j\mid n$, define $\mathcal{C}_{j,s}(n) = \{1\leq x\leq n^s (x,n^s)_s = d^s_j\}$. 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 $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ and prove that for the above linear congruence, the number of solutions is $\frac1{n^s}\sum\limits_{d\mid n}c_{d,s}(b)\prod\limits_{j=1}^{\tau(n)}\Bigl(c_{n/{d_j},s}\Bigl(\frac{n^s}{d^s}\Big)\Big)^{g_j}$ where $g_j = |\{x_1,\ldots, x_k\}\cap\mathcal{C}_{j,s}(n)|$ for $j=1,\ldots, \tau(n)$ and $c_{d,s}$ denotes the generalized Ramanujan sum defined by E. Cohen (1955). |
