Question

How many solutions does the equation 6x = 4 have in (a)${\mathbf{\mathbb{Z}}}_{\mathbf{7}}$?

Short answer

The solution of the equation is x = [3]₇.

Step-by-step solution

Write the elements

The set of${\mathrm{\mathbb{Z}}}_7$ contains the elements {0,1,2,3,4,5,6}. As the number 7 is prime, at least one solution of the given equation exits.

Find the solution

As [6]₇ = [-1]₇ and [4]₇ = [-3]₇ , so the solution of the given equation can be found by multiplying 6 by equivalence classes of 3.

This implies that there are no other solutions of the equation as ${\left[-1\right]}_7·x\ne {\left[-3\right]}_7$for $x\ne 3$.

So, this implies that:

$$\begin{gathered} 6·{\left[3\right]}_7={\left[18\right]}_7 \\ ={\left[4\right]}_7 \end{gathered}$$

The solution of the equation isx = [3]₇.