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    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
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    (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.
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    (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.
 
 
 
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