Example 13 There are three steel wires, one is 13 feet 5 inches long, one is 24 feet 3 inches long, and one is 55 feet 8 inches long. Now they need to be cut into equal small segments, with no remainder from any of the wires, and the segments should be as long as possible. What is the length of each small segment in inches? How many segments can be cut in total?
Since
$$135=3^{3} \times 5, \quad 243=3^{5}, \quad 558=2 \times 3^{2} \times 31,$$

thus $(135,243,558)=9$. Also,
$$\frac{135}{9}+\frac{243}{9}+\frac{558}{9}=15+27+62=104$$

Answer: The length of each small segment is 9 inches, and a total of 104 segments can be cut.