Here is the problem: Let be a positive integer. Show that in any set of consecutive integers there is exactly one divisible by .
Here is the solution:
Let be the integers in the sequence. The integers , , are distinct because whenever . Because there are n possible values for and there are n different integers integers in the set, each of these values is taken exactly once. It follows that there is exactly one integer in the sequence that is divisible by n.
Something I don't quite understand: How does make the values calculated from the modulus distinct? Also, what is the motivation
for ? Why are we allowed to use this fact?