How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3$? 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.

Consider the following seven numbers:
$a_1\\a_2\\a_3\\a_4\\a_5\\a_1+a_2+a_3+a_4\\a_2+a_3+a_4+a_5$
Since there are seven numbers three of them must have same remainder modulo $3$. If all are from the first five then their sum is multiple of $3$ otherwise one of the four-element-sum and one of its summand must be included and their difference is divisible by $3$.
