Question

Based on Exercise 1 and 2, make a conjecture about units and zero divisors in ${\mathrm{\mathbb{Z}}}_n$ .

Short answer

• An element $\left[a\right]\in {\mathrm{\mathbb{Z}}}_n$ is a unit if and only if $\left(a,n\right)=1$ .
• An element $\left[a\right]\in {\mathrm{\mathbb{Z}}}_n$ is a zero divisor if and only if $\left(a,n\right)>1$ .

Step-by-step solution

Make conjecture about units

Based on exercise 1, the conjecture about the unit is:

An element $\left[a\right]\in {\mathrm{\mathbb{Z}}}_n$ is a unit if and only if role="math" localid="1646382821791" $\left(a,n\right)=1$
.

Prove the conjecture as:

As role="math" localid="1646382877302" $\left(a,n\right)=1$ is equivalent to $ax+ny=1$ , where$xandy$ are integers. By Taking modulo n of the equation $ax+ny=1$, we have $\left[a\right]\left[x\right]=1$. Thus, conjecture is true.

Make conjecture about zero divisor 

Based on exercise 2, the conjecture about the unit is:

An element $\left[a\right]\in {\mathrm{\mathbb{Z}}}_n$ is a zero divisor if and only if $\left(a,n\right)>1$ .

Prove the conjecture as:

As $\left(a,n\right)>1$, this implies that:

Thus, the conjecture about zero divisors is true.