# Tour:Multiplicative monoid modulo n

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WHAT YOU NEED TO DO: Read, and make sure you understand, the definitions and examples below. Recall that a monoid is a set with associative binary operation having a two-sided identity element, but without necessarily having inverses.

## 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.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Cyclicity is subgroup-closed|UP: Introduction four (beginners)|NEXT: Multiplicative group modulo n