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Homework Help: 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


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    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


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    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

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

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