Q:

Constructing a minimum sized cube

We have  several identical rectangular prisms that has dimensions 3m width, 5m height and 6m length.  We will construct a cube and We are free to place  each cube every way we like.   How many of them must be used in order to construct the cube which has the smallest volume?  
 I know if i choose the side of the cube lcm(3，5，6) =30  then i will have to use $\frac{30*30*30}{3*5*6}$  =300 such prisms.  I am wondering if there is a way , by placing differently to get a smaller sized cube

A:

Each prism has volume $3\cdot 5\cdot 6=90=2\cdot 3^2 \cdot 5$.  So the cube that you construct must have volume that is a multiple of $90$.  The least cube multiple of $90$ is $2^3 \cdot 3^3 \cdot 5^3 = 27000$.
So yes, you will need (at least) $300$ of your prisms to make your cube.