Prove that it is divisible by 8.

[Michael_Light]

Homework Statement

Prove 9ⁿ-1 is divisible by 8 such that n is positive integer.

Homework Equations

The Attempt at a Solution

It looks simple and i tried to apply everything i know but yet i can't prove it. Any hints?

 

[dynamicsolo]

Do you know modular (clock) arithmetic? What is 9 modulo 8? What about 9²? What happens each time you multiply by another factor of 9?

(If you aren't familiar with this device, think about a clock with "hours" from 0 to 7 [eight positions]. Start at "0" and count 9 "hours" forward; where do you end up? What happens for multiples of 9? Where is 9²? What happens with higher powers of 9?

Also, where are all multiples of 8 located on the clock?)

 

[uart]

Homework Statement

Prove 9ⁿ-1 is divisible by 8 such that n is positive integer.

Homework Equations

The Attempt at a Solution

It looks simple and i tried to apply everything i know but yet i can't prove it. Any hints?

Hi Michael. It's very easy if you use induction.

Consider how you could write $(9^{n+1} - 1)$ in terms of $(9^n-1)$.