Changing signs to minus signs to obtain a sum of zero.

Question

Consider the sum $1+2+3+...+101$. Is it possible to change some of the plus signs to minus signs so that the sum is zero?

Well, I know by using Gauss' method $1+2+...+100=5050$ then $5050+101=5151$

So I started to see if I can find a pattern but Im not sure.

Here is what I did: I aligned the numbers like so:

$1+2+3+4+...+50$

$100+98+97+96+...+51$

and then we can't forget about the $101$. So I need the above to give me $-101$ in order for the sum to be zero.

I noticed if I add $101+1=101$

So I gotta change the sign of $2$ and $98$ making it $-2$ and $-98$. So every even number and its partner has to have their sign changed such as $2,4,6,8,10...50$

but how am i guaranteed that the sum will be $-101$

Answer 1

If the answer is yes then this means that we can divide these numbers on two parts one with positive sign and the other with negative sign and their sum is $0$ so the sum of these number would be even which is a contradiction.

Answer 2

No, it is not possible because $$1\pm 2\pm\cdots \pm 101\equiv 1+0+1+0+\cdots+1 \equiv 51\equiv 1\pmod 2.$$