Results 1 to 2 of 2

Math Help - x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    13

    x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

    Show that x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
    Posts
    411
    You mean: "if and only if" ?

    Anyway: look at resctlasses modulo 4

    That is: \mathbb{Z}/4\mathbb{Z} = \left\{0,1,2,3\right\}

    Let x\equiv 0 mod 4. Then x^n\equiv 0 mod 4
    Let x\equiv 1 mod 4. Then x^n\equiv 1 mod 4
    Let x\equiv 2 mod 4. Then x^n\equiv 0 mod 4 for n\geq 2.

    Let x\equiv 3 mod 4. Then x^n\equiv 1 mod 4 for even n. And x^n\equiv 3 mod 4 for odd n.

    Hence the conclusion follows.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. a ≡ b ?
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 7th 2011, 02:06 AM
  2. Replies: 0
    Last Post: June 28th 2010, 09:32 PM
  3. a^p≡1 (mod p^n) iff a≡1 (mod p^(n-1)), n≥2
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: March 8th 2010, 09:59 AM
  4. lcm of k,k+1,k+2 where k≡3(mod 4)
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: February 12th 2010, 02:46 PM
  5. x≡y (mod a) => (x,a)=(y,a)
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: February 7th 2010, 06:05 AM

Search Tags


/mathhelpforum @mathhelpforum