Question

Solve the equation.

$x^3+x^2=\left[2\right]$in${\mathrm{\mathbb{Z}}}_{10}$

Short answer

$x=\left[1\right],\left[2\right]$ are the solutions of$x^3+x^2=\left[2\right]$ in${\mathrm{\mathbb{Z}}}_{10}$ .

Step-by-step solution

Obtain the solution

Consider the given equation

$x^3+x^2=\left[2\right]$in ${\mathrm{\mathbb{Z}}}_{10}$

$x$	$x^3\oplus x^2$			Solution
$\left[0\right]$	$\left[0^3+0^2+\right]$	$=\left[0\right]$	$=\left[0\right]$	No
$\left[1\right]$	$\left[1^3+1^2+\right]$	$=\left[2\right]$	$=\left[2\right]$	Yes
$\left[2\right]$	$\left[2^3+2^2+\right]$	$=\left[12\right]$	$=\left[2\right]$	Yes
$\left[3\right]$	$\left[3^3+3^2+\right]$	$=\left[36\right]$	$=\left[6\right]$	No
$\left[4\right]$	$\left[4^3+4^2+\right]$	$=\left[80\right]$	$=\left[0\right]$	No
$\left[5\right]$	$\left[5^3+5^2+\right]$	$=\left[150\right]$	$=\left[0\right]$	No

Result

The solution of $x^3+x^2=\left[2\right]$ in ${\mathrm{\mathbb{Z}}}_{10}$ is $x=\left[1\right],\left[2\right]$.