Question

Prove that the product of two units in ${\mathrm{\mathbb{Z}}}_n$, is also a unit.

Short answer

It is proved that the product of two units is also a unit in ${\mathrm{\mathbb{Z}}}_n$.

Step-by-step solution

Use the given part

It is given that there are two units in ${\mathrm{\mathbb{Z}}}_n$.

Let the two units be $a$and $b$in ${\mathrm{\mathbb{Z}}}_n$. Then for these two units, there exists $x$and $y$in ${\mathrm{\mathbb{Z}}}_n$so that $ax=1$and $by=1$.

Proof 

Now, multiply and using the fact that the product is commutative in ${\mathrm{\mathbb{Z}}}_n$.

$\begin{array}{rcl}\left(ab\right)\left(xy\right)& =& a\left(bx\right)y\\ & =& axby\\ & =& \left(ax\right)\left(by\right)\\ & =& 1·1\\ & =& 1\end{array}$

Thisimplies that product of $a$and $b$is also unit in ${\mathrm{\mathbb{Z}}}_n$.

Hence proved.