132. For which $x$ and $y$ is the number $x x y y$ a square of a natural number?
Solution. The number $\overline{x x y y}$ can be represented as $11 \cdot \overline{x 0 y}$ $\overline{x x y y}=1000 x+100 x+10 y+y=1100 x+11 y=11(100 x+y)=11 \times$ $\times x 0 y)$. Therefore, it will be a square only in the case when $x 0 y$ is divisible by 11 and the quotient is also a square of a natural number. But $\overline{x 0 y}=100 x+y=(x+y)+99 x$, 152
so the sum $x+y$ must be divisible by 11, i.e., $x+y=11$. By simple enumeration, we get the solution: $x=7, y=4$.

Answer: the number 7744 is a square of a natural number.