Prove that it is divisible by 8.

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In summary, the conversation discusses how to prove that 9n-1 is divisible by 8 for positive integers, using modular arithmetic and induction. The hint is to consider how (9^(n+1) - 1) can be written in terms of (9^n-1).
  • #1
Michael_Light
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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?
 
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  • #2
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?)
 
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  • #3
Michael_Light said:

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 [itex](9^{n+1} - 1)[/itex] in terms of [itex](9^n-1)[/itex].
 

FAQ: Prove that it is divisible by 8.

How do you prove that a number is divisible by 8?

There are a few ways to prove that a number is divisible by 8. One way is to divide the number by 8 and see if the result is a whole number. Another way is to check if the last three digits of the number are divisible by 8. If either of these methods result in a whole number, then the original number is divisible by 8.

Can you use the prime factorization method to prove divisibility by 8?

Yes, the prime factorization method can also be used to prove that a number is divisible by 8. If the number has at least three factors of 2 in its prime factorization, then it is divisible by 8.

What is the rule for divisibility by 8?

The rule for divisibility by 8 is that a number is divisible by 8 if its last three digits are divisible by 8. In other words, the number must end in 000, 008, 016, 024, 032, etc. to be divisible by 8.

Can you use a calculator to prove divisibility by 8?

Yes, most calculators have a "mod" function which can be used to determine the remainder when dividing a number by 8. If the remainder is 0, then the number is divisible by 8.

Are all multiples of 8 divisible by 8?

Yes, by definition, a multiple of 8 is any number that can be evenly divided by 8. Therefore, all multiples of 8 are divisible by 8.

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