# 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 $n$ 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.