We need it and we do have it as I wrote it since it is not true in general that ALL the residues modulo N are coprime to N...in your example, with N = 8, the residues 0,2,4,6 are not coprime with 8.
All the NON-ZERO residues modulo N are coprime to N iff N is a prime.
And perhaps you shouldn't look for resources in the web about group theory: it's too wide and complex a subject to read about it just to solve this problem.
There must be some more elementary way to solve then this problem, but right now I can't see it.
Tonio