Prove:that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014 Show that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014 without remainder ($i\not=j$). 
I think the "pigeonholes" here is the remainder of division by 2014 . 
now i need to think of a way to define the "pigeons" but cant think of anything . 

Hint: Pigeons are the remainders when you divide $\pm a_i$ by $2014$.  Holes are the possible remainders. How many pigeons and holes are there? And what happens if two pigeons are in the same hole?
