Prove that the number of elements of is , where denotes the Euler totient function.
By a previous theorem, the distinct elements of are precisely the classes . Recall that consists precisely of those residue classes such that there exists with . If and , we have such that , and thus . In particular, and are relatively prime. Conversely, if and are relatively prime, then there exist and such that , and thus . Hence, the elements of are precisely those residue classes whose representatives in the (integer) range are relatively prime to . By definition, there are of these.