Results 1 to 7 of 7

Thread: Find all solutions in Z16

  1. #1
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128

    Find all solutions in Z16

    Find all solutions in Z16 to the equation $\displaystyle x^2 =9$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,769
    Thanks
    3027
    Quote Originally Posted by 450081592 View Post
    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!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128
    Quote Originally Posted by HallsofIvy View Post
    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,769
    Thanks
    3027
    Quote Originally Posted by 450081592 View Post
    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 $\displaystyle 10^2= 4 (mod 16)$.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128
    Quote Originally Posted by HallsofIvy View Post
    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?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by 450081592 View Post
    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
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128
    Quote Originally Posted by tonio View Post
    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?
    Last edited by 450081592; Nov 25th 2009 at 10:21 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find solutions of z bar = z^2
    Posted in the Pre-Calculus Forum
    Replies: 19
    Last Post: Aug 29th 2011, 02:50 AM
  2. Help! find all solutions to..
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: May 6th 2009, 02:40 PM
  3. find all solutions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Mar 23rd 2009, 04:26 PM
  4. Find solutions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 8th 2009, 02:20 PM
  5. Find all solutions for....
    Posted in the Trigonometry Forum
    Replies: 6
    Last Post: Dec 10th 2008, 04:07 AM

Search Tags


/mathhelpforum @mathhelpforum