Linear Congruences and a Conjecture of Bibak

We address three questions posed by Bibak KB20, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences ₈=₁ᵏ aᵢ xᵢ b n. In particular, we obtain explicit expressions for the number of solutions where xᵢ's are squares modulo n. In addition, we obtain expressions for the number of solutions with order restrictions x₁ xₖ or, with strict order restrictions x₁> > xₖ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties.