# Find all solutions in Z16

• November 24th 2009, 03:54 PM
450081592
Find all solutions in Z16
Find all solutions in Z16 to the equation $x^2 =9$.
• November 24th 2009, 06:46 PM
HallsofIvy
Quote:

Originally Posted by 450081592
Find all solutions in Z16 to the equation $x^2 =9$.

Have you even tried? If nothing else, just start squaring!

For example, what is $1^2$, what is $2^2$, etc. modulo 16? You probably could have done all 15 possible cases in less time than it took your computer to warm up!
• November 24th 2009, 09:05 PM
450081592
Quote:

Originally Posted by HallsofIvy
Have you even tried? If nothing else, just start squaring!

For example, what is $1^2$, what is $2^2$, etc. modulo 16? You probably could have done all 15 possible cases in less time than it took your computer to warm up!

um I just need some explanation for the question, I only know how to find the solution from a matrix, not a equation
• November 25th 2009, 04:08 AM
HallsofIvy
Quote:

Originally Posted by 450081592
um I just need some explanation for the question, I only know how to find the solution from a matrix, not a equation

A matrix? Just do the arithmetic!

$0^2= ?$
$1^2= ?$
$2^2= ?$
$3^2= ?$
$4^2= ?$
$5^2= ?$
$6^2= ?$
$7^2= ?$
$8^2= ?$
$9^2= ?$
$10^2= ?$
$11^2= ?$
$12^2= ?$
$13^2= ?$
$14^2= ?$
$15^2= ?$
All of those are "modulo 16" of course. For example $10^2= 100= 96+ 4= 6(16)+ 4$ so $10^2= 4 (mod 16)$.
• November 25th 2009, 08:53 AM
450081592
Quote:

Originally Posted by HallsofIvy
A matrix? Just do the arithmetic!

$0^2= ?$
$1^2= ?$
$2^2= ?$
$3^2= ?$
$4^2= ?$
$5^2= ?$
$6^2= ?$
$7^2= ?$
$8^2= ?$
$9^2= ?$
$10^2= ?$
$11^2= ?$
$12^2= ?$
$13^2= ?$
$14^2= ?$
$15^2= ?$
All of those are "modulo 16" of course. For example $10^2= 100= 96+ 4= 6(16)+ 4$ so $10^2= 4 (mod 16)$.

0 up to 15, that's it, that easy?
• November 25th 2009, 09:07 AM
tonio
Quote:

Originally Posted by 450081592
0 up to 15, that's it, that easy?

No, in fact it is easier: since $x^2=(-x)^2\!\!\!\!\pmod m$ ,for any $m$ , you only have to check half the numbers, since for example $9^2=7^2\!\!\!\!\pmod{16}\,,\,or\,\,13^2=3^2\!\!\!\ !\pmod{16}$ , etc.

Tonio
• November 25th 2009, 09:10 AM
450081592
Quote:

Originally Posted by tonio
No, in fact it is easier: since $x^2=(-x)^2\!\!\!\!\pmod m$ ,for any $m$ , you only have to check half the numbers, since for example $9^2=7^2\!\!\!\!\pmod{16}\,,\,or\,\,13^2=3^2\!\!\!\ !\pmod{16}$ , etc.

Tonio

what is the relation between 9 and 7, 13 and 3, I mean what is the pattern of the equal numbers?