# On a strange pigeonhole principle problem

Given distinct integers $a_1, a_2, \cdots, a_{63}$. Prove that there exists $a_i, a_j, a_m, a_n$ such that $(a_i - a_j)(a_m - a_n)$ is divisible by $1984$.

I have no idea of how to create the pigeonhole because normally a pigeonhole principle question is like "there exists $i,j$", but this one is about the existence of four indices which make me think of using the principle twice. Also, we have $1984=64\cdot 31$ which is probably useful, thanks in advance.

Since $2\cdot 31<63$, there are at least three numbers such that $a_i\equiv a_j\equiv a_k\pmod{31}$. Similarly, there are two numbers such that $a_m\equiv a_n\pmod{32}$. Of $a_i$, $a_j$ and $a_k$, take the ones of same parity. We are done.