# Tour:Multiplicative monoid modulo n

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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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 $n$ be a positive integer. The multiplicative monoid modulo $n$ is defined as follows:

• Its underlying set is the set $\{ 0,1,2,\dots,n-1\}$.
• The product of two elements $a,b$ in the set, is defined by the rule: multiply them as integers, and then take the remainder of the product modulo $n$.

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

## Facts

• The multiplicative monoid modulo $n$ is a monoid of size $n$.
• The multiplicative monoid modulo $n$ has identity element (neutral element) $1$ and zero element (nil element) $0$. It is not a group.
• The multiplicative monoid modulo $n$ 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.
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