Question

(a) Give three examples of equations of the form $ax=b$in ${\mathrm{\mathbb{Z}}}_{12}$that have no nonzero solutions.

(b) For each of the equations in part (a), does the equation $ax=0$have a nonzero solution?

Short answer

(a) The three equations are $2x=7$,$2x=9$ , and $2x=11$.

(b) Yes, the equation $ax=0$have a non-zero solution.

Step-by-step solution

Part (a)

Consider three equations of the form $ax=b$.

$$\begin{gathered} 2x=4 \\ 3x=6 \\ 4x=8 \end{gathered}$$

For all the equations,thesolution is,$x=\frac{1}{2}$ , which is non-zero solutions in ${\mathrm{\mathbb{Z}}}_{12}$.

But the equations, $2x=7$, $2x=9$, and $2x=11$has no non-zero solutions or zero solutions.

Part (b)

Consider three equations of the form $ax=0$.

2,3,and 4 are all zero divisors in ${\mathrm{\mathbb{Z}}}_{12}$.

Let the equations $2x=0$. it has solution a solution $x=6$.