Chapter 9.4, Problem 37ES

a. Suppose $a_1,a_2,\mathrm{...},a_n$ is a sequence of n integers none of which is divisible by n. Show that at least one of the differences $a_i-a_j$ (for $i\ne j$) must be divisible by n.
b. Show that every finite sequence $x_1,x_2,\mathrm{...},x_n$ of n integers has a consecutive subsequence $x_{i+1},x_{i+2},\mathrm{...},x_j$ 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.