Answer
Showing that in any set of n consecutive integers there is exactly one divisible by n.
Work Step by Step
--Let a, a + 1, . . . , a + n − 1 be the integers in the sequence.
- The integers (a + i) mod n, i = 0, 1, 2, . . . , n − 1, are distinct, because
0 < (a +j)−(a +k) < n whenever 0 ≤ k < j ≤ n−1.
- Because there are n possible values for (a + i) mod n and there
are n different integers in the set, each of these values is taken
on exactly once.
- It follows that there is exactly one integer in
the sequence that is divisible by n.
