Maybe this helps.An example:
n = 20;
Solve[x^10 + 2 x^3 - 20 == 0, x, Modulus -> n]
Solve[Mod[x^10 + 2 x^3 - 20, n] == 0 &&
0 < x < n, x, Integers](*x=0 only missing*)
Reduce[Mod[x^10 + 2 x^3 - 20, n] == 0 &&
0 < x < n, x, Integers](*x=0 only missing*)
And:
Table[Solve[x^10 + 2 x^3 - 20 == 0, x, Modulus -> n], {n, 2,
20}] // MatrixForm
If n=8 and n=16 is no solution.
And then yours equation:
ClearAll["`*"];
Solve[Mod[n/(n - x) - n, n] == 0 && 0 < x < n, x, Integers]
Reduce[Mod[n/(n - x) - n, n] == 0 && 0 < x < n, x, Integers]
Mathematica gives only general solution, but it dosen't know under which C1 is no solution.
Regards M.I.