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Simple Proof

  1. Feb 26, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove the following: If the integer n is divisible by 3 then n^3 is divisible by 3.

    2. Relevant equations

    Direct Proof

    3. The attempt at a solution

    n = 3m

    n^2 = 9m^2

    n^2 = 3(3m^2)

    I think the proof is done at this point because the 3 factors out but I also did this:

    n^2 = 9m^2

    (n^2)/9 = m^2

    (n/3)(n/3) = (m)(m)

    which also implies n is divisible by 3 since integer x integer = integer

    My professor is kind of harsh on proofs so Im not sure if there are intermediate steps I'm missing. Thanks!
     
  2. jcsd
  3. Feb 26, 2008 #2

    Gib Z

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    Homework Helper

    If 3 is a factor of n, it must also be a factor of any natural power of n. Do you mean n^3 or n^2? Your working looks like n^2. Anyway, the first way you did it is correct and satisfactory though if your teacher is really harsh, you may want to go like:

    [tex]n= 3m[/tex] for some natural value of m, because n is divisible by 3 (Data).
    Squaring both sides, [tex]n^2 = 9m^2 = 3 ( 3m^2)[/tex]. Since m is natural, 3m^2 must also be natural, and hence 3 is a factor of n^2 as well.
     
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