• Support PF! Buy your school textbooks, materials and every day products Here!

Proof of a-1 divides a^n-1

  • #1

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
Science Advisor
Homework Helper
41,770
911
The simplest way to do that is to observe that [tex](1)^n- 1= 0[/tex]. What does that tell you?
 
  • #3
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
Science Advisor
Homework Helper
26,258
618
Halls meant do you know the Remainder Theorem. If not then you should try to factor a^n-1. Start with n=2.
 
  • #5
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
938
9
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
Science Advisor
Homework Helper
Dearly Missed
10,705
1,722

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
 

Related Threads for: Proof of a-1 divides a^n-1

Replies
1
Views
2K
  • Last Post
Replies
3
Views
1K
Replies
9
Views
5K
  • Last Post
Replies
0
Views
1K
Replies
6
Views
3K
Replies
3
Views
594
Top