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Math Help - Even or Odd

  1. #1
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    Even or Odd

    The numbers 1, 2, 3, ..., 2007, 2008 are recorded on a sheet of paper. You may cross out any two numbers and replace them by their difference. If you repeat this operation enough times, a single number will be left. Is this number even or odd? Why?
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  2. #2
    Super Member PaulRS's Avatar
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    Quote Originally Posted by bearej50 View Post
    The numbers 1, 2, 3, ..., 2007, 2008 are recorded on a sheet of paper. You may cross out any two numbers and replace them by their difference. If you repeat this operation enough times, a single number will be left. Is this number even or odd? Why?
    Just note that given 2 integers x and y we have x-y\equiv{x+y}(\bmod.2) that is x-y and x+y have the same parity.

    Thus in our case we deduce that the final number will have the same parity as 1+2+...+2008=\frac{2008\cdot{2009}}{2} which is multiple of 2. Thus the final number is even!
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  3. #3
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    thank you tons
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