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Math Help - proof required....

  1. #1
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    proof required....

    can somebody plz find a value or show me how to find a value for x such tht,

    2^x/3 = n

    where x and n are integers

    or otherwise prove tht for no integeral value of x can 2^x be an integeral multiple of 3.
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  2. #2
    Eater of Worlds
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    An induction proof should work well with this.

    An observation to make is that powers of 2 alternate between congruencies depending if n is odd or even.

    2^{n}\equiv{1}(mod \;\ 3), if n is even

    2^{n}\equiv{-1}(mod \;\ 3), if n is odd.

    That is, 2^{n}-1 divides 3 if n is even and 2^{n}+1 divides 3 if n is odd.
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  3. #3
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    Quote Originally Posted by maliksaim View Post
    can somebody plz find a value or show me how to find a value for x such tht,

    2^x/3 = n

    where x and n are integers

    or otherwise prove tht for no integeral value of x can 2^x be an integeral multiple of 3.
    Not sure what class you are in exactly, but
    \frac{2^x}{3} = n

    2^x = 3n

    Now, if x and n are integers, then 2^x and 3n are integers.

    So look at the prime factorization of both sides. Clearly the prime factorization of the RHS contains a 3, but the prime factorization of the LHS contains only 2s. Thus the two sides cannot be equal for x and n integers.

    -Dan
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  4. #4
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    Quote Originally Posted by maliksaim View Post
    can somebody plz find a value or show me how to find a value for x such tht,

    2^x/3 = n

    .
    It is not possible we require that 3 | 2^x which is simply impossible.
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  5. #5
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    Quote Originally Posted by maliksaim View Post
    [snip]
    or otherwise prove tht for no integeral value of x can 2^x be an integeral multiple of 3.
    Quote Originally Posted by ThePerfectHacker View Post
    [snip]
    we require that 3 | 2^x which is simply impossible.
    Gets my vote for proof of the year!!
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    Gets my vote for proof of the year!!
    The question was stupid anyway.
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  7. #7
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    yea the question seemed stupid, but it was really required for something very interesting, the MIU formal system from "godel, eshcer, bach". i thought i'd give it a try. i guess i just wasnt meant to take the glory of proving it
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