What I've observed: Pick any $3$ random positive integers, say $a$, $b$, $c$ which are not of the form $0\pmod{3}$ then one and only one of $a+b$, $b+c$, $c+a$, $a+b+c$ is always a multiple of $3$.
What I've generalized: Let $a_1$$;$ $a_2$; $...;$ $a_k$ be $k$ positive integers with $a_r$ $\not=$ $0\pmod{k}$ $\forall$ $1$ $\le$ $r$ $\le$ $k$. Then there exist $m$ and $n$ with $1$ $\le$ $m$ $\le$ $n$ $\le$ $k$ such that $\sum_{i=m}^n a_i$ is divisible by $k$.
My question: Whether or not such a generalization is true.
Note: The condition $a_r$ $\not=$ $0\pmod{k}$ $\forall$ $1$ $\le$ $r$ $\le$ $k$ is given to avoid the trivial solution, it being $m$ $=$ $n$ $=$ $r$.