In the case of complete residue systems, the algebraic structure of congruences have been proved to be a commutative ring $\mathbb Z_m.$ If the integer $m$ was equal to a prime number $p$, then the ring became even a field, which was proven in this proposition. But what happens to the structure, if we consider reduced residue systems, i.e. if we allow only congruences classes co-prime to $m$? The following proposition clarifies this question.

Proposition: Multiplicative Group Modulo an Integer $(\mathbb Z_m^*,\cdot)$

Let $m > 1$ be an integer. The algebraic structure $(\mathbb Z_m^*,\cdot)$ containing the congruence classes of a reduced residue systems modulo $m$ builds with respect to the multiplication operation "$\cdot$" a commutative group called the multiplicative group modulo $m$.


Proofs: 1

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  1. Kraetzel, E.: "Studienb├╝cherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981