Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*

In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if p is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ∗ = Z/pZ\{0} is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/p∗. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ∗ and prove it is cyclic.