# Find all solutions in Z16

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

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

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

For example, what is \$\displaystyle 1^2\$, what is \$\displaystyle 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!
• Nov 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 \$\displaystyle 1^2\$, what is \$\displaystyle 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
• Nov 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!

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

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

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

0 up to 15, that's it, that easy?
• Nov 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 \$\displaystyle x^2=(-x)^2\!\!\!\!\pmod m\$ ,for any \$\displaystyle m\$ , you only have to check half the numbers, since for example \$\displaystyle 9^2=7^2\!\!\!\!\pmod{16}\,,\,or\,\,13^2=3^2\!\!\!\ !\pmod{16}\$ , etc.

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

Originally Posted by tonio
No, in fact it is easier: since \$\displaystyle x^2=(-x)^2\!\!\!\!\pmod m\$ ,for any \$\displaystyle m\$ , you only have to check half the numbers, since for example \$\displaystyle 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?