Math Help - rectangular prism problem

1. rectangular prism problem

A rectangular prism 6cm by 3cm by 3cm is made up by stacking 1cm by 1cm by 1cm cubes. How many rectangular prism, including cubes, are there whose vertices are vertices of the cube, and whose edges are parallel to the edges of the original rectangular prism? (rectangular prisms with the same dimensions but in different position are different).

2. Originally Posted by 0110kim
A rectangular prism 6cm by 3cm by 3cm is made up by stacking 1cm by 1cm by 1cm cubes. How many rectangular prism, including cubes, are there whose vertices are vertices of the cube, and whose edges are parallel to the edges of the original rectangular prism? (rectangular prisms with the same dimensions but in different position are different).
You have got $6 \cdot 3 \cdot 3 = 54$ 1-cm³-cubes.

Calculate all divisors of 54 and add them up.

$54 : \rightarrow\ \{1, 2, 3, 6, 9, 18, 27, 54 \}$

That means:
You have 1 prism containing 54 unit-cubes
You have 2 prisms containing 27 unit-cubes
You have 3 prisms containing 18 unit-cubes ... etc.

Therefore there are 120 different prisms.