IGNORE my last solution. I misread the question.
Let 'L' be the total lenght of the wire. Upon segmenting as per the requirement, each segment has
1st wire peice - L/12
2nd wire peice - L/18
3rd wire piece - L/24
The LCM for these values is 72. This is the minimum value.
To go about solving these... draw a line lenght "L". cut in to 3 peices, each of L/3. And we know these each L/3 shud be cut in to 4, 6, and 8 parts. Hence..
L/3 x 4= L/12
L/3 x 6 = L/18
L/3 x 8 = L/24.
Calculate the LCM. In this case, 72. Hence 72 is the answer. Pardon me for misunderstanding the question, earlier.
Giving another SHOT