Results 1 to 2 of 2

Thread: A divisibility problem

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    16

    A divisibility problem

    Show that if $\displaystyle (a,b)=1$ and $\displaystyle p$ is an odd prime, then

    $\displaystyle ( a+b, (a^p+b^p)/a+b ) = 1$ or $\displaystyle p$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by eyke View Post
    Show that if $\displaystyle (a,b)=1$ and $\displaystyle p$ is an odd prime, then

    $\displaystyle ( a+b, (a^p+b^p)/a+b ) = 1$ or $\displaystyle p$
    $\displaystyle \frac{a^p+b^p}{a+b}=\sum_{j=1}^p (-1)^{j-1} a^{p-j}b^{j-1}= \sum_{j=1}^p (-1)^{j-1} (a+b \ - \ b)^{p-j}b^{j-1} \equiv pb^{p-1} \mod a+b.$ similarly $\displaystyle \frac{a^p+b^p}{a+b} \equiv pa^{p-1} \mod a+b.$ so if $\displaystyle d \mid a+b$ and $\displaystyle d \mid \frac{a^p + b^p}{a+b},$ then $\displaystyle d \mid pa^{p-1}$ and $\displaystyle d \mid pb^{p-1}.$

    thus $\displaystyle d \mid p \gcd(a^{p-1},b^{p-1})=p. \ \Box$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Divisibility problem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Dec 8th 2011, 06:24 PM
  2. [SOLVED] Divisibility Problem
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Jun 28th 2011, 07:53 AM
  3. Divisibility Problem(2)
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Sep 18th 2010, 01:02 PM
  4. A divisibility problem 3
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Feb 27th 2010, 04:15 PM
  5. help this divisibility problem
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Apr 29th 2008, 05:13 AM

Search Tags


/mathhelpforum @mathhelpforum