Question: a. Suppose a1, a2,..., an is a sequence of n integers none of which is divisible by n. Show that at least one of the differences ai - aj (for i = j) must be divisible by n.
b. Show that every finite sequence x1, x2,..., xn of n integers has a consecutive subsequence xi+1, xi+2,..., xj whose sum is divisible by n.
(For instance, the sequence 3, 4, 17, 7, 16 has the consecutive subsequence 17, 7, 16 whose sum is divisible by 5.) (From: James E. Schultz and William F Burger, "An Approach to Problem Solving Using Equivalence Classes Modulo n," College Mathematics Journal (15), No. 5 1984, 401-405)