Question 70
Prove that a natural number $A$ is a perfect square if and only if, for every nonzero natural number $n$, at least one of the differences
$$
(A+1)^{2}-A,(A+2)^{2}-A,(A+3)^{2}-A, \ldots,(A+n)^{2}-A
$$
is divisible by $n$.
$$
(A+1)^{2}-A,(A+2)^{2}-A,(A+3)^{2}-A, \ldots,(A+n)^{2}-A
$$
is divisible by $n$.