Answer
The shortest-length carton which can accommodate boxes of either size without any room left is $24$ in. long.
Work Step by Step
We have to calculate the shortest-length carton which can accommodate boxes of either size without any room left.
That means the lengths of both the boxes should divide the length of carton.
To find that least common multiple.
Factorize $6$ as follows:
$6=2\cdot3$
Now factorize $8$ as follows:
$8=4\cdot2=2\cdot2\cdot2$
Now select factors of $8$.
That are, $2\cdot2\cdot2$
Since, these factors are missing $3$ which is a factor of $6$.
Multiply $3$ in the factors of $8$.
We get, $2\cdot2\cdot2\cdot3=8\cdot3=24$
We get, the least common multiple $=24$
So, the shortest-length carton which can accommodate boxes of either size without any room left is $24$ in. long.
