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Proof that if 32 -|- ((a^2+3)(a^2+7)) if a is even

  1. Sep 15, 2012 #1

    mhz

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    1. The problem statement, all variables and given/known data

    Proof that if 32 -|- ((a^2+3)(a^2+7)) if a is even. (note: -|- means NOT divides)

    The question asks for a proof of this statement.

    2. Relevant equations

    If a | b, then there exists an integer k such that b = ka.

    3. The attempt at a solution

    If the contrapositive of the statement is true, then so is the statement. So we will prove that if a is odd, then 32 | ((a^2+3)(a^2+7)).


    Since a is odd, (a^2+3)(a^2+7) is even, 2 | ((a^2+3)(a^2+7)). This must mean that there exists an integer k such that ((a^2+3)(a^2+7)) = 2k. ...
     
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  3. Sep 15, 2012 #2

    dextercioby

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    32 can't divide the product of 2 odd numbers, right ?
     
  4. Sep 15, 2012 #3

    mhz

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    Right, but I'm trying to prove that in a sort of fashion that I've already shown (long-winded and specifically showing each step, trying not to assume much).

    What you're arguing is something like:

    Suppose a is even. Then a^2 is even, so both a^2+3 and a^2+7 are odd. Since 32 is even, 32 \nmid ((a^2+3)(a^2+7)).

    However that logic is flawed. This is saying that since the converse of the statement is true, so too must the statement be true. (but that is wrong)
     
  5. Sep 15, 2012 #4

    dextercioby

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    You are in error. The proof you posted which you read from my statement is perfecty ok. There's no converse anywhere. The converse would have been something like:

    Prove that if 32 doesn't divide (a^2 +3) (a^2 +7) then a is even.

    As you can see the converse in not involved in the proof you posted.
     
  6. Sep 15, 2012 #5

    mhz

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    That is the contrapositive, not the converse.

    Converse of "if A then B" is "if B then A", the contrapositive is "if not B then not A".

    Since A in this case is "if 32 doesn't divide (a^2 +3) (a^2 +7)" and B is "a is even". The converse would be "if a is even then 32 doesn't divide (a^2 +3) (a^2 +7)" and the proof I posted in my second post proves that. However, my point is that the proof of the converse does not always prove the statement.

    What I'm trying to do is one of the following:
    a) Prove the statement: "if 32 doesn't divide (a^2 +3) (a^2 +7) then a is even"
    b) Prove the contrapositive: "if a is odd, 32 divides (a^2 +3) (a^2 +7)"

    A little while after my last post I came up with this, and I think this proof is ok:

     
  7. Sep 17, 2012 #6

    jbunniii

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    I think the confusion is stemming from the fact that you have two "ifs" in your problem statement. One of them should not be there, and the meaning will be different depending on which one you keep.
     
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