ILL 1970 Problem 12

Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$ . $(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$ , where the $\pm$ signs are independent of each other. $(b)$ Generalize the result to every prime number.