Let $x_1,x_2,x_3,x_4,x_5,x_6$ be given integers, not divisible by $7$. Prove that at least one of the expressions of the form $$\pm x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$$ is divisible by $7$, where the signs are selected in all possible ways. (Generalize the statement to every prime number greater than two!)
Each term is in $\{1,2,3,4,5,6\}$ modulo $7$, but how do I use this to prove the result?