TITLE: How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3$?
QUESTION [6 upvotes]: How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3?$
I realize that this is a number theory question and we use modular arithmetic, but I'm unsure of where to begin with this specific situation.

REPLY [4 votes]: Consider the remainders of the numbers when divided by 3. Clearly there are only three possible remainders: 0, 1 or 2.

If all these remainders occur among the given numbers, then adding together three numbers with different remainders will yield a sum that is divisible by 3 (since 0 + 1 + 2 = 3).
Otherwise, at least one of the three remainders does not occur among the five given numbers. By the pigeonhole principle, at least one of the other two remainders must thus occur more than twice (since 5 > 2 × 2). Adding together three numbers with the same remainder will then yield a sum divisible by 3.