IMO 1984 SL 10

Origin: GBR

Problem

Prove that the product of five consecutive positive integers cannot be the square of an integer.

Solution

Suppose that the product of some five consecutive numbers is a square. It is easily seen that among them at least one, say n, is divisible neither by 2 nor 3. Since n is coprime to the remaining four numbers, it is itself a square of a number m of the form 6k \pm 1. Thus n = (6k \pm 1)2 = 24r + 1, where r = k(3k \pm 1)/2. Note that neither of the numbers 24r −1, 24r + 5 is one of our five consecutive numbers because it is not a square. Hence the five numbers must be 24r, 24r + 1, . . . , 24r + 4. However, the number 24r + 4 = (6k \pm 1)2 + 3 is divisible by 6r + 1, which implies that it is a square as well. It follows that these two squares are 1 and 4, which is impossible.