Problem - 1705
Six distinct positive integers are randomly chosen between $1$ and $2020$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$?
By the pigeonhole principle, we find at at least two of them are in the same residue class MOD $5$ and, thus, their difference must be a multiple of $5$. It follows that the answer is $1$.