Even or Odd

bearej50 · #1 $Old$ 11-18-2008, 06:55 AM

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?

PaulRS · #2 $Old$ 11-18-2008, 07:07 AM

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!

bearej50 · #3 $Old$ 11-18-2008, 01:05 PM

thank you tons