Results 1 to 2 of 2

Math Help - Number Theory

  1. #1
    Junior Member
    Joined
    Apr 2008
    From
    Gainesville
    Posts
    68

    Number Theory

    Let a,b \in  \mathbb{Z} . with (a,4)=2 and (b,4)=2. find (a+b, 4) and prove that your answer is correct.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Smile

    Quote Originally Posted by JCIR View Post
    Let a,b \in  \mathbb{Z} . with (a,4)=2 and (b,4)=2. find (a+b, 4) and prove that your answer is correct.
    (a + b,\,4) = 4

    \emph{Proof: } First, note that the only possible divisors of 4 are 1, 2, and 4, so we only need show that 4\mid(a + b)

    (a,\,4) = 2\text{ and }(b,\,4)=2\Rightarrow2\mid a\text{ and }2\mid b\Rightarrow\exists p,\,q\in\mathbb{Z},\;a = 2p,\,b = 2q

    \Rightarrow a + b = 2(p + q)

    But 2\nmid p, for if so, \exists m\in\mathbb{Z},\;a = 2(2m) = 4m\Rightarrow4\mid a and so (a,\,4) = 4 and not 2 as required. Similarly, 2\nmid q.

    Thus p and q are odd so their sum p + q must be even (i.e., \exists s,\,t\in\mathbb{Z},\;p = 2s + 1\text{ and }q = 2t + 1 \Rightarrow p + q = 2s + 2t + 2 = 2(s + t + 1) with s + t + 1\in\mathbb{Z}). So 4\mid2(p + q)\Rightarrow4\mid(a + b) as required to show that (a + b,\,4) = 4\quad\square
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Textbooks on Galois Theory and Algebraic Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 8th 2011, 07:09 PM
  2. Number Theory
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: May 19th 2010, 08:51 PM
  3. Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 16th 2010, 06:05 PM
  4. Replies: 2
    Last Post: December 18th 2008, 06:28 PM
  5. Number theory, prime number
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 17th 2006, 09:11 PM

Search Tags


/mathhelpforum @mathhelpforum