I am currently preparing to take an entrance exam for a university in my country, which will take place soon. I recently encountered a math problem, and I am not sure if I solved it correctly. I need your advice to determine if my procedure is correct or not.
This is the exercise:
Using the smallest possible number of identical cubes, with edge size expressed naturally in centimeters, can we construct a cuboid with dimensions of 8, 12, and 28 cm?
To find the smallest number of identical cubes, with the size of the edge in centimeters expressed by a natural number, that can be assembled into a cuboid with dimensions of 8, 12, and 28 cm, we need to find the common factors of these three numbers and then multiply them together. The greatest common factor of 8, 12, and 28 is 4.
$8= 2^3$
$12= 2^2.3$
$28= 2^2.7$
Therefore, we can divide each of these numbers by 4 to get smaller numbers with no common factors.
$8 \text{cm} . 12 \text{cm} . 28\text{cm}= 2688\text{cm^2}$
$\frac{8}{4}. \text{cm} \frac{12}{4} \text{cm} . \frac{28}{4} \text{cm} = \frac{2688}{64} \text{cm}$
$2 \text{cm} . 3 \text{cm} . 7 \text{cm} = 42 \text{cm}$
Thus, we need 42 identical cubes with 2 cm x 3 cm x 7 cm dimensions to assemble a cuboid with 8 cm, 12 cm, and 28 cm dimensions.
Is this correct?