(solution) Let n 2 be an integer. Let S be a set of n + 1 integers.

(solution) Let n 2 be an integer. Let S be a set of n + 1 integers.

Let n ? 2 be an integer.

? Let S be a set of n + 1 integers. Prove that S contains two elements whose di?erence is divisible by n.

Hint: Use the Pigeonhole Principle.

? Prove that there is an integer that is divisible by n and whose decimal representation only contains the digits 0 and 5

. Hint: Consider the integers 5, 55, 555, 5555, …