I am looking at the following number theory problem: I am given any $m+2$ integers, and must show there exist two of them whose sum (or difference) is divisible by $2m$.
My first observation was that if any two of these integers are equal, their difference is 0 and hence trivially divisible by $2m$. Moreover, I take that for any two distinct integers, their difference will be divisible by $2m$ iff they have the same remainder (when divided by $2m$).
Then it should follow that given the proposition we wish to show, if the remainders of the $m+2$ integers are distinct, there exist two numbers whose sum (but not difference) is divisible by $2m$.
So to restate the problem, from a set of $m+2$ integers each with distinct remainders, how can I show there are two whose sum is divisible by $2m$?