10. (10 points) A stick of length $L$, is divided into 8, 12, and 18 equal segments by red, blue, and black lines respectively. The stick is then cut at each division line. How many segments can be obtained in total? What is the length of the shortest segment?
10. (10 points) A wooden stick of length $L$ is divided into 8, 12, and 18 equal segments by red, blue, and black lines, respectively. The stick is then cut at each division line. How many segments can be obtained in total? What is the length of the shortest segment?
【Analysis】To meet the conditions, $L$ must be a multiple of 8, 12, and 18. Therefore, first find the least common multiple (LCM) of these three numbers and the LCM of each pair, to determine the overlapping segments. Then, use the principle of inclusion-exclusion to solve the problem.
【Solution】Solution: Assume $L=[8,12,18]=72$ times $K$, i.e., $L=72K$. Then:
The red line divides the stick into 8 equal parts (9 division points), each part being $9K$ in length;
The blue line divides the stick into 12 equal parts (13 division points), each part being $6K$ in length;
The black line divides the stick into 18 equal parts (19 division points), each part being $4K$ in length;
Also, $[9K, 6K]=18K$, overlapping 4 segments; $[6K, 4K]=12K$, overlapping 6 segments; $[9K, 4K]=36K$, overlapping 2 segments;
$[9K, 6K, 4K]=36K$, overlapping 2 segments.
By the principle of inclusion-exclusion, the total number of segments is: $(8+12+18)-4-6-2+2=28$ (segments);
Or the total number of points is: $(9+13+19)-5-7-3+3=29$ (division points), so there are 28 segments in total.
Thus, the shortest segment is the distance between the red and black lines: $L \div 72=\frac{L}{72}$.