The multiplicative inverse of 1/2 (mod 108) is 2.
This is actually related to a problem in this thread, but is so much more basic I didn't want to interrupt any discussion that the original thread might generate.
The problem is to solve
Since the details of solving the problem are irrelevent here I will simply state a pair of solutions to the quadratic:
So the solutions mod 108 are 10 and 95, which both work in the original equation.
My question about all this. 1/2 has no multiplicative inverse mod 108. So even though,say, (1/2)(-3 + 23) = (1/2)(20) = (1/2)(2*10) = (1/2)*2*10 is apparently equal to 10, how can we say (1/2)*2 = 1?
WolframAlpha and I agree that the equation 2x = 1 (mod 108) has no solution!
Edit: I suppose you could say that this question is more Philosophically based than it is practical.
... so that it is 'all right'...