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Proof by contradiction, please help

  1. Feb 12, 2012 #1
    1. The problem statement, all variables and given/known data
    If a,b and c are integers and a2+b2=c2, then at least one of a and b is even.


    2. Contradiction statement
    There exist an integer a,b,c such that a2+b2=c2 and a or b is odd



    3. The attempt at a solution
    I am not sure if my contradiction statement is correct because of this part, then at least one of a and b is even. I think it means 1 has to be even and the other could be even or odd. Let me know if my contradiction is right or wrong and point me in the right direction to start.
     
  2. jcsd
  3. Feb 12, 2012 #2

    Mark44

    Staff: Mentor

    The statement "at least one of a and b is even" is true if either a or b is even, or if both a and b are even. The opposite of this statement is "neither a nor b is even."
     
  4. Feb 12, 2012 #3
    So either a or b is odd would be correct?
     
  5. Feb 12, 2012 #4

    vela

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    Suppose a=1 and b=2. Then the statement "either a or b is odd" is true because a is odd. The statement "at least one of a and b is even" is true because b is even. So "either a or b is odd" can't be the opposite of "at least one of a and b is even."
     
  6. Feb 12, 2012 #5

    Mark44

    Staff: Mentor

    If you'll read more closely, you'll see that I said neither a nor b is odd.
     
  7. Feb 12, 2012 #6
    I reread your statement and it says both a and b are not even so they both must be odd?
     
  8. Feb 12, 2012 #7

    Dick

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    Yes, to prove the contradiction you would assume they are both odd. Think about mod 4.
     
    Last edited: Feb 12, 2012
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