Question

Solve the equation.

$x^3\oplus x^2\oplus x\oplus \left[1\right]=\left[0\right]$in${\mathrm{\mathbb{Z}}}_8$

Short answer

$x=\left[3\right]$is the only solution of $x^3\oplus x^2\oplus x\oplus \left[1\right]=\left[0\right]$ in${\mathrm{\mathbb{Z}}}_8$ .

Step-by-step solution

Obtain solution

Consider the given equation

$x^3\oplus x^2\oplus x\oplus \left[1\right]=\left[0\right]$

$x$	$x^3\oplus x^2\oplus x\oplus \left[1\right]$			Solution
$\left[0\right]$	$\left[0^3+0^2+0+1\right]$	$=\left[1\right]$	$=\left[1\right]$	No
$\left[1\right]$	$\left[1^3+1^2+1+1\right]$	$=\left[4\right]$	$=\left[4\right]$	No
$\left[2\right]$	$\left[2^3+2^2+2+1\right]$	$=\left[15\right]$	$=\left[7\right]$	No
$\left[3\right]$	$\left[3^3+3^2+3+1\right]$	$=\left[40\right]$	$=\left[0\right]$	Yes
$\left[4\right]$	$\left[4^3+4^2+4+1\right]$	$=\left[85\right]$	$=\left[5\right]$	No
$\left[5\right]$	$\left[5^3+5^2+5+1\right]$	$=\left[156\right]$	$=\left[4\right]$	No

Result

The solution of$x^3\oplus x^2\oplus x\oplus \left[1\right]=\left[0\right]$ in${\mathrm{\mathbb{Z}}}_8$ is $x=\left[3\right]$ .