The question says that the wire was divided into three equal parts which in turn were divided into 4,6 and 8 parts respectively.
Now,can i say that each part after first division(after the original wire was cut) is divisible by 4,6 and 8.
Therefore to find the minimum length of the original wire...we have to find the minimum length of each equal part(after the original wire was cut) such that it is divisible by 4,6 and 8.....which is nothing but the l.c.m of 4,6 and 8.....whicih is 24.
So the minimum original length of the wire is 24*3=72.
The problem with this approach is 48 is also divisible by 48. and as we can see 6 is not a divisor of 16(one of the 3 equal part of 48).