Question

How do you find the greatest common factor of \[36x{{y}^{3}},24{{y}^{2}}\]?

Answer 1

Hint: The greatest common factor is two or more monomials is the product of all their common factors. It is the factor that divides a given number of terms. In order to find the greatest common factor of these two monomial terms, we write them in factor form of the prime numbers and then find the factor that is common in both the terms.

Formula used:
In order to find the prime factors, we start by dividing the number by the least prime number \[2\] and divide until we get a remainder. After that, we divide it by \[3,5,7\] and so on until we are left with prime numbers only .Secondly, we write all the prime numbers in the multiplication form.
For example for \[20\], we start by dividing \[20\] by \[2\] , and after dividing twice, we get \[5\] as remainder which is also a prime number so we write, \[\left( 2\times 2\times 5 \right)\].

Complete step by step solution:
Firstly, to find the greatest common factor of \[36x{{y}^{3}},24{{y}^{2}}\], write the prime factors form these two monomial terms.
For the first term, i.e.\[36x{{y}^{3}}\]
The prime factors of this term can be written as\[\Rightarrow 2\times 2\times 3\times 3\times x\times y\times y\times y\]
Now for the second term, i.e. \[24{{y}^{2}}\]
The prime factors of this term can be written as \[\Rightarrow 2\times 2\times 2\times 3\times y\times y\]
For both the terms, the prime factors that are common are:
\[\begin{align}
  & \Rightarrow 2\times 2\times 3\times 3\times x\times y\times y\times y \\
 & \Rightarrow 2\times 2\times 2\times 3\times y\times y \\
 & ~i.e. \\
 & 2\times 2\times 3\times y\times y \\
 & \Rightarrow 12{{y}^{2}} \\
\end{align}\]

Therefore, the greatest common factor is \[12{{y}^{2}}\].

Note: In this question, always start with the lowest prime factor only. Starting with a factor other than the lowest one may be troublesome. Writing \[\left( 3\times 12 \right)=36\] is factorization and not the prime factorization