Example 3. Try to find the four-digit number $\overline{x x y y}$, such that it is a perfect square.
Solution: Since the required square number $\overline{x x y y}=11 \cdot(100 x + y)$, therefore, $100 x+y$ can be divided by 11, i.e., $11 | (x + y)$. Noting that $x$ and $y$ are both digits, thus $x+y=11$. After verification, only $x=7, y=4$ satisfies the condition, hence the required four-digit square number is 7744.