1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of a-1 divides a^n-1

  1. Sep 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that if a is in Z (if a is an integer), then for every positive integer n, a-1 divides a^n -1.


    2. Relevant equations



    3. The attempt at a solution
    I'm really not entirely sure where to start with this one. Can someone help?
     
  2. jcsd
  3. Sep 19, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The simplest way to do that is to observe that [tex](1)^n- 1= 0[/tex]. What does that tell you?
     
  4. Sep 19, 2012 #3
    Wouldn't this not work if a=1 then? Because then a -1 = 1 -1 = 0 and a^n - 1 = 1^n - 1 = 1 - 1 = 0. So you would always be trying to divide 0 by 0, which is undefined.
     
  5. Sep 19, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Halls meant do you know the Remainder Theorem. If not then you should try to factor a^n-1. Start with n=2.
     
  6. Sep 19, 2012 #5
    Oh! Okay! Thanks!
     
  7. Sep 20, 2012 #6
    Actually it's even simpler than that. What does it mean that a=1 is always the solution to an-1 = 0?
     
  8. Sep 20, 2012 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Induction on n is another (easy) way to go.

    RGV
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of a-1 divides a^n-1
Loading...