a. Suppose
a
1
,
a
2
,
...
,
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
≠
j
) must be divisible by n.
b. Show that every finite sequence
x
1
,
x
2
,
...
,
x
n
of n integers has a consecutive subsequence
x
i
+
1
,
x
i
+
2
,
...
,
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.