Question

Quentin has 18 unit cubes. how many different rectangular prims can he build if he uses all of the cubes ?

Answer 1

A rectangular prism is defined by three lengths.

We can find out how many unit cubes would be in a prism by multiplying these three lengths together--that's how we find our volume.

Similarly, we can come up with different ways to multiply together three different numbers and make 18.
Each combination would be a new rectangular prism, with one catch:
Order doesn't matter. A prism with lengths 2, 2, and 3 is the same as one with lengths 2, 3, and 2, so don't make that mistake.

To find each combination, keep splitting 18 in different ways.
If one of the ways we split it can also be split, we need to write out that, too.
Here are the possible combinations:
18 × 1 × 1, obviously
9 × 2 × 1. splitting off 2
6 × 3 × 1. splitting off 3
4 × 6 × 1. our next biggest we can take out is 6, which can also be split...
4 × 3 × 2. there's the split of 6 into 2 and 3
(3 × 6 × 1 is a repeat.)
3 × 3 × 2 is new, though
(2 × 9 × 1 is a repeat...)
(2 × 3 × 3 is a repeat...)
(aaaand 1 × 1 × 18 is a repeat. let's count up our combinations.)


There are 6 possible ways to multiply numbers together and get 18...
So 6 possible rectangular prisms.