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

## The Attempt at a Solution

I'm really not entirely sure where to start with this one. Can someone help?

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

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

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?

Ray Vickson
Homework Helper
Dearly Missed

## 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.

## 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