For linear congruences of the form

Let gcd(a, c) = d. I know that if d = 1, then there is one unique class of solutions.

If , there there is no solutions.

However, I did a problem in the book and the answer said that there are three classes of solutions:

By the Euclidean Algorithm

9 = 3*3 + 0

Thus, (3, 9) = 3 | 6,then there are three solutions. This part I do not understand.

It follows by solving the Diophantine equation we get , where and

So, then , ,

Is there a theorem that states how many solutions there are? From the problem, I believe that the number of solutions is equal to the gcd.