MORE GENERAL. Prove in every set of $n$ different numbers that are bigger or equal to zero there is at least one subset that if you sum up its elements you get a number is a multiple of $n$.
Proof. Let $a_1,a_2, \cdots, a_n$ be the $n$ integers. Consider $n$ sums: $S_1=a_1, S_2=a_1+a_2, S_3=a_1+a_2+a_3, \cdots, S_n=a_1+a_2+ \cdots a_n$ Case 1. If in this $n$ sums, there is at least one sums is divisible by $n$. Q.E.D.
Case 2. If in this $n$ sums, there is no sums which is divisible by $n$. Hence, by Pigeonhole principle, there is exist $2$ sums $S_i,S_j \; (1 \le i such that $n|S_j-S_i$ or $n|a_{i+1}+a_{i+2}+ \cdots+ a_j$. We have Q.E.D.