Chapter 2.6, Problem 10E

Solve each of the following equations by finding ${\left[a\right]}^{-1}$ and using the result in Exercise $9$.

$\begin{array}{llll}a\text{.}\left[\text{4}\right]\left[x\right]\text{ = }\left[\text{5}\right]\text{ in }{\mathbb{Z}}_{13}\hfill & \hfill & \hfill & b\text{. }\left[\text{8}\right]\left[x\right]\text{ = }\left[\text{7}\right]\text{ in }{\mathbb{Z}}_{11}\hfill \\ c\text{. }\left[\text{7}\right]\left[x\right]\text{ = }\left[11\right]\text{ in }{\mathbb{Z}}_{12}\hfill & \hfill & \hfill & d\text{. }\left[\text{8}\right]\left[x\right]\text{ = }\left[\text{11}\right]\text{ in }{\mathbb{Z}}_{15}\hfill \\ e\text{. }\left[\text{9}\right]\left[x\right]\text{ = }\left[\text{14}\right]\text{ in }{\mathbb{Z}}_{20}\hfill & \hfill & \hfill & f\text{. }\left[\text{8}\right]\left[x\right]\text{ = }\left[\text{15}\right]\text{ in }{\mathbb{Z}}_{27}\hfill \\ g\text{.}\left[\text{6}\right]\left[x\right]\text{ = }\left[\text{5}\right]\text{ in }{\mathbb{Z}}_{319}\hfill & \hfill & \hfill & h\text{. }\left[\text{9}\right]\left[x\right]\text{ = }\left[\text{8}\right]\text{ in }{\mathbb{Z}}_{242}\hfill \end{array}$

Let $\left[a\right]$ be an element of ${\mathbb{Z}}_n$ that has a multiplicative inverse ${\left[a\right]}^{-1}$ in ${\mathbb{Z}}_n$. Prove that $\left[x\right]={\left[a\right]}^{-1}\left[b\right]$ is the unique solution in ${\mathbb{Z}}_n$ to the equation $\left[a\right]\left[x\right]=\left[b\right]$.