Multiplicative monoid modulo n

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.