Question

Prove that every nonzero element of ${\mathrm{\mathbb{Z}}}_n$. is either a unit or a zero divisor, but not both.

Short answer

It is proved that is a zero divisor.

Step-by-step solution

Use the given part

Assume that is a non-zero element of ${\mathrm{\mathbb{Z}}}_n$, then there exists some integer $x$, such that $ax=1$does not have any solution. Hence,we have to show that $a$isazero divisor.

Show that a is zero divisor

As $ax=1$in ${\mathrm{\mathbb{Z}}}_n$, then by division algorithm, it can be written as $ax=qn+1$, where $q$is any integer quotient. Then we get

$ax-qn=1$

Thisimplies $\left(a,n\right)=1$.

From part (a),$ax=1$ does not have any solution, which means $\left(a,n\right)\ne 1$, so let $\left(a,n\right)=d$, then:

$a·\frac{n}{d}=0$

It is known that n divides d, and the result is less then n, where$\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$d\in $}\right.{\mathrm{\mathbb{Z}}}_n$ .Hence,$a$ is a zero divisor.