Does a-1 Always Divide a^n-1 for Any Integer a?

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Homework Help Overview

The discussion revolves around proving that for any integer \( a \), \( a-1 \) divides \( a^n - 1 \) for every positive integer \( n \). This involves concepts from number theory and polynomial division.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore starting points for the proof, including the Remainder Theorem and factoring \( a^n - 1 \). There are questions about the implications when \( a = 1 \) and concerns about division by zero.

Discussion Status

Participants are actively discussing various approaches, including induction and specific cases. Some guidance has been offered regarding factoring and the Remainder Theorem, but no consensus has been reached on a definitive method.

Contextual Notes

There is an emphasis on the conditions under which the statement holds, particularly regarding the case when \( a = 1 \) and the implications of dividing by zero.

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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?
 
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The simplest way to do that is to observe that (1)^n- 1= 0. What does that tell you?
 
HallsofIvy said:
The simplest way to do that is to observe that (1)^n- 1= 0. 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.
 
Halls meant do you know the Remainder Theorem. If not then you should try to factor a^n-1. Start with n=2.
 
Dick said:
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!
 
Actually it's even simpler than that. What does it mean that a=1 is always the solution to an-1 = 0?
 
eaglemath15 said:

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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