Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a restricted linear congruence   Let $b,n\in \mathbb{Z}$, $n\geq 1$, and ${\cal D}_1, \ldots, {\cal
D}_{\tau(n)}$ be all positive divisors of $n$. For $1\leq l \leq \tau(n)$,
define ${\cal C}_l:=\lbrace 1 \leqslant x\leqslant n \; : \; (x,n)={\cal
D}_l\rbrace$. In this paper, by combining ideas from the finite Fourier
transform of arithmetic functions and Ramanujan sums, we give a short proof for
the following result: the number of solutions of the linear congruence
$x_1+\cdots +x_k\equiv b \pmod{n}$, with $\kappa_{l}=|\lbrace x_1, \ldots, x_k
\rbrace \cap {\cal C}_l|$, $1\leq l \leq \tau(n)$, is \begin{align*}
\frac{1}{n}\mathlarger{\sum}_{d\, \mid \,
n}c_{d}(b)\mathlarger{\prod}_{l=1}^{\tau(n)}\left(c_{\frac{n}{{\cal
D}_l}}(d)\right)^{\kappa_{l}}, \end{align*} where $c_{d}(b)$ is a Ramanujan
sum. Some special cases and other forms of this problem have been already
studied by several authors. The problem has recently found very interesting
applications in number theory, combinatorics, computer science, and
cryptography. The above explicit formula generalizes the main results of
several papers, for example, the main result of the paper by Sander and Sander
[J. Number Theory {\bf 133} (2013), 705--718], one of the main results of the
paper by Sander [J. Number Theory {\bf 129} (2009), 2260--2266], and also gives
an equivalent formula for the main result of the paper by Sun and Yang [Int. J.
Number Theory {\bf 10} (2014), 1355--1363].

32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph