Question

How many solutions does the equation 6x = 4 have in ${\mathbf{\mathbb{Z}}}_{\mathbf{9}}$?

Short answer

The given equation has no solution in ${\mathrm{\mathbb{Z}}}_9$.

Step-by-step solution

Write the elements

The set of${\mathrm{\mathbb{Z}}}_9$ contains the elements $\left\{0,1,2,3,4,5,6,7,8\right\}$.

Find the solution

Directly substitute the value into the equation asfollows:

$$\begin{gathered} 6·{\left[1\right]}_9={\left[6\right]}_9 \\ 6·{\left[2\right]}_9={\left[12\right]}_9 \\ ={\left[3\right]}_9 \\ 6·{\left[3\right]}_9={\left[18\right]}_9 \\ ={\left[0\right]}_9 \end{gathered}$$

It is observed all the values of the unique multiple of 6 are in ${\mathrm{\mathbb{Z}}}_9$. This implies that the given equation has no solution in ${\mathrm{\mathbb{Z}}}_9$.

So, the given equation has no solution in ${\mathrm{\mathbb{Z}}}_9$.