So far, all I can figure out is that a and b are units (mod m). Also, a is the inverse of b, and b is the inverse of a. Say o(a) = c and o(b) = d. I want to show that c = d. I am unsure on how to do this.
I also know:
a^c is congruent to 1 (mod m)
b^d is congruent to 1 (mod m)
ab is congruent to 1 (mod m)
...and I'm lost from here on.