Question

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

Answer 1

The correct answer is:

He can make four different rectangular prisms using these cubes.

Step-by-step explanation:

We use the factors of 18 to answer this question. We want three factors of 18 at a time. To find these factors, we use our divisibility rules.

Every number is divisible by 1. 18/1 = 18; we can have a rectangular prism with dimensions 1 x 1 x 18.

Since 18 is even, it is divisible by 2: 18/2 = 9. 9 = 3 x 3; this gives us 2 x 3 x 3 for the prism.

Using 2 again, we also have 2 x 9 x 1.

The digits of 18 are 1 and 8; the sum of these is 9. Since this is divisible by 3, then 18 is divisible by 3; 18/3 = 6. This gives us 3 x 6 x 1 for the prism.

Answer 2

18 unit cubes means that the area of the rectangular prisms (cuboids), will be 18 cm^3
We need to work out the three factors that can be multiplied to get 18. Let's start simple and give the cuboid a height of one. That means we need to work out heights and lengths that are factors of 18. Factors of 18 are:

1*18
2*9
3*6

So there we have three cuboids, of dimensions 1*1*18, 1*2*9, and 1*3*6
Now things get more complicated. We need to use the other factors of 18 as heights. Let's start with a height of 2. This means that the height and length of the rectangle need to be factors of 9, because 18/2 = 9.
Factors of 9 are:

1*9
3*3

So that gives us two cuboids, of dimensions 2*1*9 and 2*3*3
Use the next factor, which is 3. 18/3 = 6, so the height and length have to be factors of 6. Factors of 6 are:

1*6
2*3

So that gives us two cuboids, of dimensions 3*1*6 and 3*2*3
At this point, we don't need to go through any more factors, because they will all certainly end up with the same dimensions as the previous ones we've found

Our final cuboids are 1*1*18, 1*2*9, 1*3*6, 2*1*9, 2*3*3, 3*1*6, 3*2*3
Now we need to eliminate all the duplicates.
1*2*9 = 2*1*9
1*3*6 = 3*1*6
2*3*3 = 3*2*3
So we'll just get rid of all the ones on the right hand side, because they're equivalents of the ones on the left
So we end up with cuboids of dimensions 1*1*18, 1*2*9, 1*3*6, 2*3*3

Hope that helps, I know some of that may be confusing, you can comment/pm and I'll try to explain it more :)