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

[eaglemath15]

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?

 

[HallsofIvy]

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

 

[eaglemath15]

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.

 

[Dick]

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

 

[eaglemath15]

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!

 

[clamtrox]

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

 

[Ray Vickson]

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