11.4 The eccentric has $N$ unit squares, from which he managed to form a rectangle with sides differing by 9. The eccentric did not stop there and formed another rectangle from the same $N$ squares, but this time with sides differing by 6. Find $N$.
Solution 1: If we denote the smaller sides of the constructed rectangles by $x$ and $y$, we get the equation $N=x(x+9)=y(y+6)$. Multiply both sides by 4.

$$
\begin{gathered}
4 x^{2}+36 x=4 y^{2}+24 y \\
(2 x+9)^{2}-81=(2 y+6)^{2}-36 \\
(2 x+2 y+15)(2 x-2 y+3)=45
\end{gathered}
$$

Both brackets on the left side are integers, and the left one is strictly greater than 15. The only divisor of the number 45 that is greater than 15 is the number 45 itself. Therefore, $2 x+2 y+15=45, 2 x-2 y+3=1$. Solving the system, we find $x=7, y=8$. Hence, $N=7 \cdot 16$.

Solution 2: Notice that if $x \geqslant y$, then the left side is at least $y(y+9)>y(y+6)$, and if $y \geqslant x+2$, then the right side is at least $(x+2)(x+8)=x^{2}+10 x+16>x^{2}+9 x=x(x+9)$. Therefore, $y=x+1$, from which it is not difficult to obtain $x=7$ and $y=8$.

## Criteria:

- When checking divisors, only natural divisors are considered, and the positivity of the two factors is not obvious - up to 2 points are deducted;
- The answer contains extraneous solutions with non-positive $x$ and $y$ - 2 points are deducted.