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

Jeremy Wood

in Mathematics

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Whitney Matthews on April 1, 2018

18 unit cubes means that the area of the rectangular prisms (cuboids), will be 18 cm^3We need to work with the three factors that can be multiplied to get 18. We're going to start simple and give the cuboid a height of one. That means we have to work heights, and lengths that are factors of 18 years. Factors of 18 are:1*182*93*6So there you have three aspects, dimensions 1*1*18, 1*2*9, and 1*3*6Now things get more complicated. We have to use the other factors of 18 as the heights. Let's start with a height of 2. This means that the height and length of the rectangle must be factors of 9, because 18/2 = 9.Factors of 9:1*93*3So that gives us two cuboids of dimensions 2*1*9 and 2*3*3Use the next factor, which is 3. 18/3 = 6, so that the height and length have to be factors of 6. Factors of 6 are:1*62*3So that gives us two cuboids of dimensions 3*1*6 and 3*2*3at this point, we don't need to go through more factors, because all certainly end up with the same dimensions than the previous ones that we have foundOur end cuboids are 1*1*18, 1*2*9, 1*3*6, 2*1*9, 2*3*3, 3*1*6, 3*2*3Now we need to delete all the duplicates. 1*2*9 = 2*1*91*3*6 = 3*1*62*3*3 = 3*2*3So let's get rid of all who are on the right side, because they are equivalent to the leftSo we ended up with aspects of dimensions 1*1*18, 1*2*9, 1*3*6, 2*3*3Hope that helps, I know some of that can be confusing, you can comment/pm and I'll try to explain it more :)


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