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?
Thanks!
-Dan
I really don't want to dispute this because it obviously works, but both WolframAlpha and I agree that the equation 2x = 1 (mod 108) has no solution!
-Dan
Edit: I suppose you could say that this question is more Philosophically based than it is practical.
The 'division' in the field of integers mod 108 is performed as 'ordinary division' and that means that n must be an even number. The equation You have written is solvable only if is a 'perfect square' mod 108 and is even. In Your case is...
... so that it is 'all right'...
Kind regards