Solving a system of linear equations modulo n

I have a system of linear equations

$$a+b+c \equiv 31 \pmod{54}$$ $$4a+2b+c \equiv 3 \pmod{54}$$ $$9a+3b+c \equiv 11 \pmod{54}$$

What should I input (I'm using LinearSolve)? It doesn't seem to work for composite modulo numbers. I have been unable to get Mathematica to give me all the possible solutions.

Answer

Working with LinearSolve we encounter some inconsistency of the related option Modulus -> z if z is not prime. Nonetheless we could do this

Mod[ LinearSolve[ {{1, 1, 1}, {4, 2, 1}, {9, 3, 1}}, {31, 3, 11}], 54]

{18, 26, 41}

Unfortunately we can get only one solution unlike when working with Solve. These posts describe another problems or bugs related to Modulus or Mod:

Solving/Reducing equations in Z/pZ
Strange behaviour of Reduce for Mod[x,1]

Note that the latter points some bugs present in versions 7 and 8 which have been fixed in version 9 of Mathematica.

Even though LinearSolve doesn't appear to be an appropriate approach w can use the Modulus option in another equation-solving functionality like e.g.: Solve or Reduce and in some other functions related to algebraic manipulations. This yields a symbolic result:

Solve[{  a + b + c   == 31, 
       4 a + 2 b + c ==  3,
       9 a + 3 b + c == 11}, {a, b, c}, Modulus -> 54]

{{a -> 18 + 27 C[1], b -> 26 + 27 C[1], c -> 41}}

To get a full list of solutions we should put the result in a table, (changing generated parameters since they are protected to another ones e.g. k). We can see that we need the table of length two only otherwise we would get may duplicates.

Table[ Mod[{a, b, c} /. %, 54] /. C[1] -> k, {k, 2}]

{{{45, 53, 41}}, {{18, 26, 41}}}

These are all solutions of the related system $\mod 54$:

Apply[{ Mod[#1 + #2 + #3, 54] - 31, 
        Mod[4 #1 + 2 #2 + #3, 54] - 3, 
        Mod[9 #1 + 3 #2 + #3, 54] - 11}&, %, {2}]

  {{{0, 0, 0}}, {{0, 0, 0}}}