Let $a_1,a_2,a_3,$ and $a_4$ be integers. Show that the product $\prod_{1 \leq i<j \leq 4}(a_i-a_j)$ is divisible by $12$
$\prod_{1 \leq i<j \leq 4}(a_i-a_j)=(a_1-a_2)(a_1-a_3)(a_1-a_4)(a_2-a_3)(a_2-a_4)(a_3-a_4)$ The hint is to consider the pigeonhole principle but I'm not seeing how to assign my boxes or pigeons. I know that the product of $4$ consecutive integers would be divisible by $24$ and also by $12$ but is there a way to apply the pigeonhole principle as well?
Could I consider my boxses as the remainders of $12$ and then my pigeons as all the combinations of subtracting those numbers and multiplying them to get a factor of $12$. So if the difference of any of those numbers is divisible by $12$ we are done or if multiplying any of those differences is divisible by $24$ we are done. Find some way to get more pigeons then boxses this way?
Among any four numbers, at least two of them must leave the same remainder on division by 3 (since there are only three remainders possible and we have 4 numbers). Thus the difference is a multiple of 3. Among four numbers if two leave the same remainder when divided by 4 we are done. Otherwise they all leave different remainders. Thus they must be 0,1,2,3. Thus we have two even numbers among their differences. Thus the product is also a multiple of 4.