# Proof of a-1 divides a^n-1

1. Sep 19, 2012

### eaglemath15

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. Sep 19, 2012

### HallsofIvy

The simplest way to do that is to observe that $$(1)^n- 1= 0$$. What does that tell you?

3. Sep 19, 2012

### eaglemath15

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. Sep 19, 2012

### Dick

Halls meant do you know the Remainder Theorem. If not then you should try to factor a^n-1. Start with n=2.

5. Sep 19, 2012

### eaglemath15

Oh! Okay! Thanks!

6. Sep 20, 2012

### clamtrox

Actually it's even simpler than that. What does it mean that a=1 is always the solution to an-1 = 0?

7. Sep 20, 2012

### Ray Vickson

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

RGV