Question

Solve the equation.$x^2\oplus x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_4$

Short answer

$x=\left[0\right],\left[3\right]$in${\mathrm{\mathbb{Z}}}_4$

Step-by-step solution

Solving the equation

Consider the given equation

$x^2\oplus x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_4$

$x$	$x^2$	$\oplus$	$x$			Solution
$\left[0\right]$	$\left[0\right]$	$⊝$	$\left[0\right]$	$\oplus$	$\left[0\right]=0$	Yes
$\left[1\right]$	$\left[1\right]$	$⊝$	$\left[1\right]$	$\oplus$	$\left[1\right]=2$	No
$\left[2\right]$	$\left[2\right]$	$⊝$	$\left[2\right]$	$\oplus$	$\left[2\right]=2$	No
$\left[3\right]$	$\left[3\right]$	$⊝$	$\left[3\right]$	$\oplus$	$\left[3\right]=0$	Yes

Solution of Equation

From the above table, the solution of the equation is$x=\left[0\right],\left[3\right]$in${\mathrm{\mathbb{Z}}}_4$