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 $B C=23$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f5010c49868bbc23ccb1g-26.jpg?height=431&width=488&top_left_y=1058&top_left_x=750)
Answer: 552

Solution: Let $D E=a$ and $E C=b$. Then the area of the rectangles $S=23(a+b)$. 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 $S=\frac{23 S}{k}+\frac{23 S}{l}$. Therefore, $k l=23(k+l)=>$ $k l-23 k-23 l+529=529=>(k-23)(l-23)=529$. Since $l<k$, that is, $l-23<k-23$. Therefore, $l-23=1$ and $k-23=529$. Thus, $k=D G=552$.