Question

Without using Exercises 13 and 14, prove: If $a,b\in {\mathrm{\mathbb{Z}}}_n$and is a unit, then the equation has a unique solution in ${\mathrm{\mathbb{Z}}}_n$.

Short answer

It is proved that there is a unique solution to the equation $ax=b$.

Step-by-step solution

Find the solution

It is given that $a,b\in {\mathrm{\mathbb{Z}}}_n$. Now let there exist some $a^{\text{'}}\in {\mathrm{\mathbb{Z}}}_n$so that $a^{\text{'}}a=a{a}^{\text{'}}=1$.

Then we can have $a^{\text{'}}=\left(ab\right)=\left(a{a}^{\text{'}}\right)b=b$.

Hence, is a solution to equation $ax=b$.

Show that the solution is unique 

Now assume that there any other solution to the equation $ax=b$.Let the solution is $c$;then equation will of the form $ac=b$.

Multiply both the sides of role="math" localid="1646393884626" $ac=b$by $a^{\text{'}}$.

$$\begin{gathered} a^{\text{'}}ac=a^{\text{'}}b \\ \to c=a^{\text{'}}b \end{gathered}$$

This show that the solution is the same. Hence, there is a unique solution to the equation $ax=b$.