Restricted linear congruences, by Khodakhast Bibak and Bruce M. Kapron and Venkatesh Srinivasan and Roberto Tauraso and L\'aszl\'o T\'oth

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. Some special cases of this problem have been already studied in many papers. The problem is very well-motivated and in addition to number theory has intriguing applications in combinatorics, computer science, and cryptography, among other areas.