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If a and b are positive odd integers, then prove that one of (a+b)/2 and (a-b)/2 are even or odd number. I mean in the above numbers one is odd and one is an even number.

  • Math - ,

    Let a>b
    a=2n+1+2k ,n,k is positive
    If k is odd,
    (a+b)/2 is even
    (a-b)/2 is odd

    If k is even,
    (a+b)/2 is odd
    (a-b)/2 is even

    The same method also applies if a<b
    (However I assume that a,b are distinct number because if a=b, (a-b)/2=0 but 0 is neither odd nor even)

  • Math - to black_widow - ,

    excuse me? 0 is an even number. The definition of even is that it leaves a zero remainder when divided by 2.

  • Math - ,

    sorry about that....
    I thought 0 is neither even nor odd....

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