# 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 result:

n =  60

#### Solution:

$a=12 \ \text{dm} \ \\ b=16 \ \text{dm} \ \\ c=20 \ \text{dm} \ \\ \ \\ 12=2^2 \cdot 3 \ \\ 16=2^4 \ \\ 20=2^2 \cdot 5 \ \\ \text{GCD}(12, 16, 20)=2^2=4 \ \\ \ \\ x=GCD(a,b,c)=GCD(12,16,20)=4 \ \text{dm} \ \\ \ \\ V_{1}=a \cdot \ b \cdot \ c=12 \cdot \ 16 \cdot \ 20=3840 \ \text{dm}^3 \ \\ V_{2}=x^3=4^3=64 \ \text{dm}^3 \ \\ \ \\ n=V_{1}/V_{2}=3840/64=60$

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