Question

Solve the following equations:

1. $x\oplus x\oplus x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_3$
2. $x\oplus x\oplus x\oplus x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_4$
3. $x\oplus x\oplus x\oplus x\oplus x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_5$

Short answer

1. Every element of ${\mathrm{\mathbb{Z}}}_3$is the solution.
2. Every element of${\mathrm{\mathbb{Z}}}_4$ is the solution.
3. Every element of${\mathrm{\mathbb{Z}}}_5$ is the solution.

Step-by-step solution

Solve the equation x⊕x⊕x=0

Apply the definition of addition in${\mathrm{\mathbb{Z}}}_n$ , that is,$\left[a\right]\oplus \left[c\right]=\left[a+c\right]$ into$x\oplus x\oplus x$ asfollows:

$\begin{array}{rcl}x\oplus x\oplus x& =& \left[x+x+x\right]\\ & =& \left[3x\right]\end{array}$

We can say that$\overline{)3}3x$ for any value of$x$ , this implies that the possible solution of the given equation is$x=\left[0\right],\left[1\right],\left[2\right]$ . We can say that every element of ${\mathrm{\mathbb{Z}}}_3$is the solution of the given equation.

Solve the equation x⊕x⊕x⊕x=0

Apply the definition of addition in ${\mathrm{\mathbb{Z}}}_n$, that is, $\left[a\right]\oplus \left[c\right]=\left[a+c\right]$into $x\oplus x\oplus x\oplus x$as follows:

$\begin{array}{rcl}x\oplus x\oplus x\oplus x& =& \left[x+x+x+x\right]\\ & =& \left[4x\right]\end{array}$

We can say that$\overline{)4}4x$ for any value of$x$ , this implies that the possible solution of the given equation is$x=\left[0\right],\left[1\right],\left[2\right],\left[3\right]$ . We can say that every element of${\mathrm{\mathbb{Z}}}_4$ is the solution of the given equation.

Solve the equation x⊕x⊕x⊕x⊕x=0

Apply the definition of addition in ${\mathrm{\mathbb{Z}}}_n$, that is, $\left[a\right]\oplus \left[c\right]=\left[a+c\right]$into $x\oplus x\oplus x\oplus x\oplus x$as follows:

$\begin{array}{rcl}x\oplus x\oplus x\oplus x\oplus x& =& \left[x+x+x+x+x\right]\\ & =& \left[5x\right]\end{array}$

We can say that $\overline{)5}5x$for any value of $x$, this implies that the possible solution of the given equation is$x=\left[0\right],\left[1\right],\left[2\right],\left[3\right],\left[4\right]$. We can say that every element of ${\mathrm{\mathbb{Z}}}_5$is the solution of the given equation.