Question

Chloe has $20$ units cubes. How many different rectangular prisms can she build with cubes?

Answer 1

Hint: In solving the above given question we will use factorization method. In mathematics, factorization or factoring consists of writing a number as a product of several factors, usually smaller or simpler objects of the same kind. In the above we have given that the number $n$ is equal to $20$. By using the factorization method we get by how many ways she can arrange boxes.

Complete step by step solution:
We have been given the value of $n$is $20$ in the above question. The way of finding the factor is very easy. Just think of those numbers which can divide $20$ without leaving any remainder. So the numbers which can divide $20$ are 1, 2, 4, 5, 10 and 20. These are the factors we required.
Now to find the ways of arrangement what we will do, we will pick out any two numbers from these factors 1, 2, 4, 5, 10, and 20 and we will determine if there is any third value from these factors so that the product of the three factors is$20$.
Here we get 4 combinations of numbers that will work:
$\begin{align}
  & \Rightarrow 1\times 1\times 20 \\
 & \Rightarrow 1\times 2\times 10 \\
 & \Rightarrow 1\times 4\times 5 \\
 & \Rightarrow 2\times 2\times 5 \\
\end{align}$
 Hence By using these four ways the Chloe can build different rectangular prisms with $20$units cubes.

Note:
There is also a quickest method to find the factors of any number. The fastest method to find the factor is to divide it by the smallest prime number (bigger than 1) that goes into it evenly with no remainder. Continue this process with each number you get, until you reach 1. Like here we have 20, first we will divide it by 2 we get 10 and no remainder left then by 5 we get 4 and no remainder left now only 1 is left hence our process is completed.