In ax $\equiv$ 1 (mod m) , when gcd(a, m) = 1, there is exactly one solution, i.e., when it exists, a modular multiplicative inverse is unique.
This is written in wikipedia. I am confused because i keep on thinking if a = 2 and m = 3. Then x can be 2,5,8 etc. I know i am missing something but i am unable to get it. I am on my own and thus would be happy if someone can clear my doubt.