You are Here: Home >< Maths

Why cant 2^n be divisible by 3? C3 HELP PLEASE?! watch

1. Question 7 on the OCR MEI C3 past paper (June 2016 ) goes as follows:

- You are given that n is a positive integer. By expressing x^(2n)-1 as a product of two factors, prove that 2^(2n)-1 is divisible by 3.

So what I did was:

- x^(2n)-1 = (x^(n)-1) (x^(n)+1)
- therefore 2^(2n)-1 = (2^(n)-1) (2^(n)+1)
- I recognise that (2^(n)-1) and (2^(n)+1) are, along with 2^(n), 3 consecutive numbers. So one of the 3 is divisible by 3. However, the mark scheme states that 2^(n) cannot be divisible by 3 but why is that? Surely it's possible that neither (2^(n)-1) nor (2^(n)+1) is divisible by 3 but 2^(n) is making the proof not possible?

If anyone could explain why 2^n cannot be divided by 3 it would be greatly appreciated
2. (Original post by aaabattery)
Question 7 on the OCR MEI C3 past paper (June 2016 ) goes as follows:

- You are given that n is a positive integer. By expressing x^(2n)-1 as a product of two factors, prove that 2^(2n)-1 is divisible by 3.

So what I did was:

- x^(2n)-1 = (x^(n)-1) (x^(n)+1)
- therefore 2^(2n)-1 = (2^(n)-1) (2^(n)+1)
- I recognise that (2^(n)-1) and (2^(n)+1) are, along with 2^(n), 3 consecutive numbers. So one of the 3 is divisible by 3. However, the mark scheme states that 2^(n) cannot be divisible by 3 but why is that? Surely it's possible that neither (2^(n)-1) nor (2^(n)+1) is divisible by 3 but 2^(n) is making the proof not possible?

If anyone could explain why 2^n cannot be divided by 3 it would be greatly appreciated
The prime factors of 2^n are 2 only so 2^n cannot be a multiple of 3.

E.g. 2^4 = 2 x 2 x 2 x 2 which is clearly not a multiple of 3.

Then consider (2^n)-1, 2^n and (2^n)+1.

These are three consecutive numbers and for any three consecutive numbers one of them has to be a multiple of 3

e.g. 8,9,10 or 22,23,24

Does that make sense?

Since you know 2^n isn't a multiple of 3, it must be either 2^(n)-1 or 2^(n)+1 which is the multiple of 3.
3. (Original post by Notnek)
The prime factors of 2^n are 2 only so 2^n cannot be a multiple of 3.

E.g. 2^4 = 2 x 2 x 2 x 2 which is clearly not a multiple of 3.

Then consider (2^n)-1, 2^n and (2^n)+1.

These are three consecutive numbers and for any three consecutive numbers one of them has to be a multiple of 3

e.g. 8,9,10 or 22,23,24

Does that make sense?

Since you know 2^n isn't a multiple of 3, it must be either 2^(n)-1 or 2^(n)+1 which is the multiple of 3.
Yes! Its very clear now, I think I was overcomplicating things. Thanks a lot.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 8, 2018
Today on TSR

Congratulations to Harry and Meghan!

But did you bother to watch?

Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants