Results 1 to 3 of 3

Math Help - let a be a solution of x^2\equiv 1 (mod m). Show that m-a is also a solution.

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    let a be a solution of x^2\equiv 1 (mod m). Show that m-a is also a solution.

    let a be a solution of x^2\equiv 1 \ \mbox{(mod m)}. Show that m-a is also a solution.

    a^2\equiv 1 \ \mbox{(mod m)}

    (m-a)^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+a^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+1\equiv 1 \ \mbox{(mod m)}

    \rightarrow m-2a\equiv 0 \ \mbox{(mod 1)}

    Not sure if this is going anywhere.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by dwsmith View Post
    let a be a solution of x^2\equiv 1 \ \mbox{(mod m)}. Show that m-a is also a solution.

    a^2\equiv 1 \ \mbox{(mod m)}

    (m-a)^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+a^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+1\equiv 1 \ \mbox{(mod m)}

    \rightarrow m-2a\equiv 0 \ \mbox{(mod 1)}

    Not sure if this is going anywhere.
    m-a is congruent to -a (mod m). Square this and you get a^2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by dwsmith View Post
    let a be a solution of x^2\equiv 1 \ \mbox{(mod m)}. Show that m-a is also a solution.

    a^2\equiv 1 \ \mbox{(mod m)}

    (m-a)^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+a^2\equiv 1 \ \mbox{(mod m)}\rightarrow m^2-2am+1\equiv 1 \ \mbox{(mod m)}

    \rightarrow m-2a\equiv 0 \ \mbox{(mod 1)}

    Not sure if this is going anywhere.
    I prefer undefined's method more, but if you want to solve this the way you were, then  (m-a)^2=m^2-2am+a^2\equiv a^2\bmod{m}

    Now, we know what  a^2 is equivalent to modulo  m .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: October 22nd 2011, 12:43 PM
  2. show that the equation has only one solution
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: October 31st 2010, 08:27 PM
  3. Show that f(x)=g(x) has a unique solution
    Posted in the Calculus Forum
    Replies: 8
    Last Post: January 18th 2010, 02:44 PM
  4. d^2y/dx^2 + b^2y=0 Show y=asinbx is a solution.
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: September 24th 2009, 02:55 AM
  5. solution please...can anyone show the proof...
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: July 19th 2006, 08:12 PM

Search Tags


/mathhelpforum @mathhelpforum