Thread: Find all solutions in Z16

1. Find all solutions in Z16

Find all solutions in Z16 to the equation $x^2 =9$.

2. 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!

3. 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

4. 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 $10^2= 4 (mod 16)$.

5. 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?

6. 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

7. 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?