Linear congruences and a conjecture of Bibak

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences \(\sum_{i=1}^k=a_{i}x_{i}\equiv b\) $$\sum _{i=1}^k=a_i{x}_i\equiv b$$ (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 x₁ ⩾ … ⩾ x_k or with strict order restrictions x₁ > … > 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.