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N2 has the form 3k or 3k+1 for some integer k

  1. Aug 16, 2010 #1
    1. The problem statement, all variables and given/known data

    n2 has the form 3k or 3k+1 for some integer k

    2. Relevant equations

    n/a

    3. The attempt at a solution

    i've tried to table it for me to see, but still i don't have the general idea, should i show n2 is either divisable by 3k or 3k+1? if yes i still don't know owho
     
  2. jcsd
  3. Aug 16, 2010 #2

    HallsofIvy

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    Re: integer

    Any integer, divided by 3, has remainder 0, 1, or 2, thus any number can be written as "3k" (remainder 0), "3k+ 1" (remainder 1), or "3k+ 2" (remainder 2).

    Now try squaring those:

    [itex](3k)^2= 9k^2= 3(3k^2)[/itex], "3 times an integer" and so is of the first kind above.

    [itex](3k+ 1)^2= 9k^2+ 6k+ 1= 3(3k^2+ 2k)+ 1[/itex], "3 times an integer plus 1" and so of the second kind above.

    [itex](3k+ 2)^2= 9k^2+ 12k+ 4[/itex][itex]= 9k^2+ 12k+ 3+ 1= 3(3k^2+ 4k+ 1)+ 1[/itex] which is again of the form "3 times an integer plus 1".

    Note that none of those three forms, 3k, 3k+1, and 3k+ 2 (and every integer can be written in one of those forms), have a square of the form 3n+ 2.
     
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