Cube cuboid minimum

Let us have a cube whose edge length is expressed in centimeters and is a natural number. What is the smallest number of such identical cubes that can be made into a cuboid with dimensions of 24 cm, 32 cm, and 60 cm? How long will the edge of these cubes be?

Final Answer:

n =  720
a =  4 cm

Step-by-step explanation:

$$\begin{gathered} x=24\text{cm} \\ y=32\text{cm} \\ z=60\text{cm} \\ 24=2^3\cdot 3 \\ 32=2^5 \\ 60=2^2\cdot 3\cdot 5 \\ \text{GCD}(24,32,60)=2^2=4 \\ a=GCD(x,y,z)=GCD(24,32,60)=4\text{cm} \\ {n}_1=x/a=24/4=6 \\ {n}_2=y/a=32/4=8 \\ {n}_3=z/a=60/4=15 \\ n=n_1\cdot n_2\cdot n_3=6\cdot 8\cdot 15=720 \end{gathered}$$

Try another example