Proof of a-1 divides a^n-1

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  • #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?
 
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  • #2
The simplest way to do that is to observe that [tex](1)^n- 1= 0[/tex]. What does that tell you?
 
  • #3
HallsofIvy said:
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
Halls meant do you know the Remainder Theorem. If not then you should try to factor a^n-1. Start with n=2.
 
  • #5
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!
 
  • #6
Actually it's even simpler than that. What does it mean that a=1 is always the solution to an-1 = 0?
 
  • #7
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
 

1. What is "Proof of a-1 divides a^n-1"?

"Proof of a-1 divides a^n-1" is a mathematical concept that proves that if a and n are positive integers, then (a-1) divides evenly into (a^n-1).

2. Why is this proof important?

This proof is important because it helps establish a connection between the divisibility of a number and its exponent. It also has applications in number theory and algebraic equations.

3. What is the significance of a-1 in this proof?

The significance of a-1 in this proof is that it is a factor of a^n-1, meaning it evenly divides into a^n-1 without leaving a remainder. This is essential in proving the divisibility of a^n-1 by (a-1).

4. How is this proof typically presented?

This proof is usually presented using mathematical notation and logical reasoning. It may involve algebraic manipulations, induction, or other mathematical techniques to show that (a^n-1) can be expressed as (a-1) multiplied by another integer.

5. Are there any exceptions to this proof?

Yes, there are exceptions to this proof. It only applies when both a and n are positive integers. If a or n is a negative integer, then the proof does not hold. Additionally, if a=1 or n=1, then (a-1) would be equal to 0, making the proof invalid.

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