Respuesta :
Answer:
The solutions of the given equation are:
x=0,1,4 and 9
Step-by-step explanation:
We are asked to find the solution of the equation:
[tex]x^2-x=0\ \text{mod}\ 12[/tex]
i.e. we have to find the possible values of x such that the equation is true.
- If x=0
then
[tex]x^2-x=0-0\\\\i.e.\\\\x^2-x=0[/tex]
Hence, x=0 is the solution of the equation.
- if x=1
then
[tex]x^2=1\\\\Hence,\\\\x^2-x=1-1\\\\i.e.\\\\x^2-x=0[/tex]
Hence, x=1 is a solution.
- If x=2
then
[tex]x^2=4[/tex]
i.e.
[tex]x^2-x=4-2\\\\i.e.\\\\x^2-x=2\neq 0[/tex]
Hence, x=2 is not a solution.
- If x=3
then
[tex]x^2=9[/tex]
i.e.
[tex]x^2-x=9-3\\\\i.e.\\\\x^2-x=6\neq 0[/tex]
Hence, x=3 is not a solution.
- If x=4
then
[tex]x^2=16=4\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=4-4\\\\i.e.\\\\x^2-x=0[/tex]
Hence, x=4 is a solution to the equation.
- If x=5
then
[tex]x^2=25=1\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=1-4\\\\i.e.\\\\x^2-x=-3=9\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=9\neq 0[/tex]
Hence, x=5 is not a solution.
- If x=6
then
[tex]x^2=36\\\\i.e.\\\\x^2=0\ \text{mod}\ 12\\\\i.e.\\\\x^2=0[/tex]
Hence,
[tex]x^2-x=0-6\\\\i.e.\\\\x^2-x=-6=6 \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]
Hence, x=6 is not a solution
- If x=7
then,
[tex]x^2=49=1\ \text{mod}\ 12\\\\i.e.\\\\x^2=1[/tex]
Hence,
[tex]x^2-x=1-7\\\\i.e.\\\\x^2-x=-6=6\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]
Hence, x=7 is not a solution.
- If x=8
then,
[tex]x^2=64=4\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=4-8\\\\i.e.\\\\x^2-x=-4=8\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=8\neq 0[/tex]
Hence, x=8 is not a solution.
- If x=9
then,
[tex]x^2=81=9\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2=9[/tex]
Hence,
[tex]x^2-x=9-9\\\\i.e.\\\\x^2-x=0[/tex]
Hence, x=9 is a solution.
- If x=10
then,
[tex]x^2=100=4\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=4-10\\\\i.e.\\\\x^2-x=-6=6\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]
Hence, x=10 is not a solution.
- If x=11
then,
[tex]x^2=121=1\ \text{mod}\ 12[/tex]
i.e.
[tex]x^2-x=1-11\\\\x^2-x=-10=2\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=2\neq 0[/tex]
Hence, x=11 is not a solution.
