Here is a statement I encounter: For any positive integer $n$, and $2n-1$ integers, there must exist $n$ integers in it such that the sum of these $n$ integers is a multiple of $n$.
I have no ideas how to prove this. If I have proved that the statement is true when $n$ is a prime number, how do I prove the case when $n$ is an arbitrary positive integer? Or could anybody give some useful hints about the proof of the statement?
Thank you very much!