Identical cubes

From the smallest number of identical cubes whose edge length is expressed by a natural number, can we build a block with dimensions 12dm x 16dm x 20dm?

Correct answer:

n =  60

Step-by-step explanation:

$$\begin{gathered} a=12\text{ dm } \\ b=16\text{ dm } \\ c=20\text{ dm } \\ 12=2^2\cdot 3 \\ 16=2^4 \\ 20=2^2\cdot 5 \\ \text{GCD}(12,16,20)=2^2=4 \\ x=GCD(a,b,c)=GCD(12,16,20)=4\text{ dm } \\ {V}_1=a\cdot b\cdot c=12\cdot 16\cdot 20=3840{\text{dm}}^3 \\ {V}_2=x^3=4^3=64{\text{dm}}^3 \\ n=V_1\mathrm{/}{V}_2=3840\mathrm{/}64=60 \end{gathered}$$

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