Small and big box

Boxes with dimensions of 6 cm, 10 cm, and 15 cm should fit into a cube-shaped box. What are the smallest dimensions a box can have?
How many boxes of the given dimensions can fit in it?

Final Answer:

x =  30 cm
n =  30

Step-by-step explanation:

$$\begin{gathered} a=6\text{cm} \\ b=10\text{cm} \\ c=15\text{cm} \\ 6=2\cdot 3 \\ 10=2\cdot 5 \\ 15=3\cdot 5 \\ LCM(6,10,15)=2\cdot 3\cdot 5=30 \\ x=LCM(a,b,c)=LCM(6,10,15)=30\text{cm} \end{gathered}$$

$$\begin{gathered} V_1=a\cdot b\cdot c=6\cdot 10\cdot 15=900{\text{cm}}^3 \\ {V}_2=x^3=30^3=27000{\text{cm}}^3 \\ n=V_2/V_1=27000/900=30 \end{gathered}$$

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