Question

How many solutions does the equation $6x=4$ have in ${\mathbf{\mathbb{Z}}}_{\mathbf{10}}$?

Short answer

The solutions of the equation are x = [4]₁₀ and x = [9]₁₀.

Step-by-step solution

Write the elements

The set of ${\mathrm{\mathbb{Z}}}_{10}$contains the elements $\left\{0,1,2,3,4,5,6,7,8,9\right\}$.

Find the solution

Directly substitute the value into the equation asfollows:

$$\begin{gathered} 6·{\left[1\right]}_{10}={\left[6\right]}_{10} \\ 6·{\left[2\right]}_{10}={\left[2\right]}_{10} \\ 6·{\left[3\right]}_{10}={\left[18\right]}_{10} \\ ={\left[8\right]}_{10} \\ 6·{\left[4\right]}_{10}={\left[24\right]}_{10} \\ ={\left[4\right]}_{10} \\ 6·{\left[5\right]}_{10}={\left[30\right]}_{10} \\ ={\left[0\right]}_{10} \end{gathered}$$

This implies that 10/5 = 2.

Therefore, there are two solutionsforthe given equation.

The remaining solution is obtained by adding [5]₁₀ to the solution [4]₁₀.

This implies that:

$$\begin{gathered} 6·{\left[9\right]}_{10}={\left[54\right]}_{10} \\ ={\left[4\right]}_{10} \end{gathered}$$

So, the solutions of the equation arex = [4]₁₀ and x = [9]₁₀.