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

$\displaystyle x^2 + 3x - 22 \equiv 0~\text{mod(108)}$

Since the details of solving the problem are irrelevent here I will simply state a pair of solutions to the quadratic:

$\displaystyle x \equiv -\frac{3}{2} \pm \frac{1}{2} \cdot 23 \equiv \frac{1}{2} ( -3 \pm 23 ) ~\text{(mod 108)}$

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