5. (8 points) Rectangles $A B C D, D E F G, C E I H$ have equal areas and integer sides. Find $D G$, if $B C=13$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f5010c49868bbc23ccb1g-38.jpg?height=429&width=488&top_left_y=1059&top_left_x=750)
Answer: 182

Solution: Let $D E=a$ and $E C=b$. Then the area of the rectangles $S=13(a+b)$. According to the condition, $S$ is divisible by $a$ and $b$, that is, $S=a k$ and $S=b l$, where $k=D G$ and $l=C H-$ are natural numbers. Then $a=\frac{S}{k}$ and $b=\frac{S}{l}$. We get that $S=\frac{13 S}{k}+\frac{13 S}{l}$. Therefore, $k l=13(k+l)=>$ $k l-13 k-13 l+169=169=>(k-13)(l-13)=169$. Since $l<k$, that is, $l-13<k-13$. Therefore, $l-13=1$ and $k-13=169$. Thus, $k=D G=182$.