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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- Nov 18th 2008, 04:55 AM #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?

- Nov 18th 2008, 05:07 AM #2
Just note that given 2 integers $\displaystyle x$ and $\displaystyle y$ we have $\displaystyle x-y\equiv{x+y}(\bmod.2)$ that is $\displaystyle x-y$ and $\displaystyle x+y$ have the same parity.

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

- Nov 18th 2008, 11:05 AM #3