Linear Congruences and a Conjecture of Bibak

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences ${\sum }_{i=1}^k{a}_i{x}_i\equiv b\text{Mod}n$. In particular, we obtain explicit expressions for the number of solutions where $x_i$’s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1\ge \cdots \ge x_k$ or, with strict order restrictions $x_1>\cdots >x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties.