Find the number of solutions to x2 1 mod 108Solution108274 x

Find the number of solutions to x^2 1 (mod 108).

108=27*4

x^2=1 mod 108

So,

x^2=1 mod 27

x^2=1 mod 4

Hence,

x^2=1 mod 9

x=1, 8 mod 9

x=9m+1,9m-1

(9m+1)^2=81m^2+18m+1=18m+1 =1 mod 27

18m=0 mod 27

Hence, m=0 mod 3

So, x=27n+1,27n-1

x^2=1 mod 4

(27n+1)^2=(3n+1)^2=9n^2+6n+1=1 mod 4

9n^2+6n=0 mod 4

Hence, n=0 mod 2

x=54k+1,54k-1

So infinitely many solutions

If we are though looking for solution in the set {0,1,2,..,107}

Solutions are:1,55,,53,107