CZ-SK-MO 2002 Problem 3

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$ , at least one of the differences $$(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A$$ is divisible by $n$ .