Inverse: Since gcd( a , n )=1 for every a ∈ Z^*_n from Extended-Euclid ( a , n ) we obtain x and y such that ax + ny =1 ⇒ ax ≡ 1 mod n ⇒ x is the inverse of a .

Clearly both +_n and *_n are associative and commutative. Thus we have established the following theorem:

Theorem 1: Both ( Z _{n ,} +_n ) and (Z^*_{n ,} *_n ) form finite Abelian groups.

|Z^*_{n ,}| = Φ( n ) where Φ( n ) is the Euler phi function .

From unique factorization theorem n can be expressed in terms of prime factors as follows:

n = p₁^e1 p₂^e2... p_k^ek

In our example n =15

15 =3*5

Φ(15)=15(1-1/3)(1-1/5) =8

For n = 45 = 3² *5 we have Φ(45) = 45 (1-1/3)(1-1/5)=24. Thus the group ( Z^*_{45 ,} *₄₅ ) contains |Z^*₄₅ | =24 elements.