A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.
Answer: $n=324$.

Let $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\frac{a}{c}=\frac{b}{d}$, or $a d=b c$. Denote $u=\operatorname{gcd}(a, c)$ ja $v=\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have

$$
c d-a b=u v\left(y^{2}-x^{2}\right)=u v(y-x)(y+x)=76=2^{2} \cdot 19 .
$$

Since $y-x$ and $y+x$ are positive integers of the same parity and $\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. Finally we have $n=a b=x^{2} u v=324$.