# Multiplicative monoid modulo n

From Groupprops

## Definition

Let be a positive integer. The **multiplicative monoid modulo ** is defined as follows:

- Its underlying set is the set .
- The product of two elements in the set, is defined by the rule: multiply them as integers, and then take the remainder of the product modulo .

Alternatively, the multiplicative monoid modulo can be defined as the monoid of *congruence classes* mod under multiplication.

## Facts

- The multiplicative monoid modulo is a monoid of size .
- The multiplicative monoid modulo has identity element (neutral element) and zero element (nil element) . It is
*not*a group. - The multiplicative monoid modulo is Abelian: any two elements in it commute.