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Math Help - Three proofs about primes and threes

  1. #1
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    Three proofs about primes and threes

    I thought this class would be fun but it turned out to be not the thing for me. It's too late to drop the course so I'm stuck doing homework that I don't understand.

    Any help with any of these would be appreciated.

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  2. #2
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    Hello qtpipi
    Quote Originally Posted by qtpipi View Post
    I thought this class would be fun but it turned out to be not the thing for me. It's too late to drop the course so I'm stuck doing homework that I don't understand.

    Any help with any of these would be appreciated.

    (a) Suppose that n is represented by the sequence of digits a_na_{n-1}...a_1, and that s = a_n + a_{n-1}+...+a_1

    Then n = 10^{n-1}a_n + 10^{n-2}a_{n-1} + ... + a_1

    \Rightarrow n-s = (10^{n-1}-1)a_n +(10^{n-2}-1)a_{n-1} + ... +(1-1)a_1

    Now 10 = 1 \mod 9 \Rightarrow 10^i = 1 \mod 9,\forall i \in \mathbb{N}

    \Rightarrow 10^i-1 = 0 \mod 9

    \Rightarrow n-s = 0 \mod 9

    \Rightarrow n and s leave the same remainder when divided by 9

    \Rightarrow 3|n \iff 3|s (and of course, 9|n \iff 9|s as well).

    (b) In any three consecutive integers, (x-1), x, (x+1), one is always a multiple of 3. And (x-1) and (x+1) are primes, and (x-1) \ne 3 \Rightarrow neither (x-1) nor (x+1) is a multiple of 3

    \Rightarrow 3|x

    And, of course, x is even whenever (x-1) is a prime > 2.

    Hence 6|x

    (c) Consider the remainders when (x-1) and (x+1) are divided by 3, and then use the results of (a).

    Grandad

    Last edited by Grandad; April 10th 2009 at 12:34 AM. Reason: Added (b) and (c)
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