Proof of a-1 divides a^n-1

  • #1
eaglemath15
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Homework Statement


Prove that if a is in Z (if a is an integer), then for every positive integer n, a-1 divides a^n -1.


Homework Equations





The Attempt at a Solution


I'm really not entirely sure where to start with this one. Can someone help?
 

Answers and Replies

  • #2
HallsofIvy
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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?
 
  • #3
eaglemath15
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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?

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.
 
  • #4
Dick
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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.
 
  • #5
eaglemath15
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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.

Oh! Okay! Thanks!
 
  • #6
clamtrox
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Actually it's even simpler than that. What does it mean that a=1 is always the solution to an-1 = 0?
 
  • #7
Ray Vickson
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Homework Statement


Prove that if a is in Z (if a is an integer), then for every positive integer n, a-1 divides a^n -1.


Homework Equations





The Attempt at a Solution


I'm really not entirely sure where to start with this one. Can someone help?

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

RGV
 

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