Question

How do you find the LCM of 96 and 108?

Answer 1

Hint: Least common multiple (LCM) or lowest common denominator (LCD) can be calculated in two ways:
> Multiplying the prime factors with the highest exponent factor.
> The LCM formula calculation of the greatest common factor (GCF).

Complete step-by-step answer:
In this question, we want to find the LCM of 96 and 108.
For finding LCM we will use the method of multiplying the prime factors with the highest exponent factor.
First, we will calculate the prime factors of 96 and 108.
Prime factors of 96:
$ \Rightarrow 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3$
The factorization of 96 in exponential form is equal to,
$ \Rightarrow 96 = {2^5} \times {3^1}$
Prime factors of 108:
$ \Rightarrow 108 = 2 \times 2 \times 3 \times 3 \times 3$
The factorization of 108 in exponential form is equal to,
$ \Rightarrow 108 = {2^2} \times {3^3}$
Now, let us multiply the highest exponent prime factors to calculate the LCM of 96 and 108.
Here, the highest exponent of 2 is 5, and the highest exponent of 3 is 3.
$ \Rightarrow LCM\left( {96,108} \right) = {2^5} \times {3^3}$
Here, the value of ${2^5}$ is 32, and the value of ${3^3}$ is 27.
Therefore,
$ \Rightarrow LCM\left( {96,108} \right) = 32 \times 27$
Let us apply multiplication on the right-hand side.
That is equal to.
$ \Rightarrow LCM\left( {96,108} \right) = 864$

Hence, the LCM of 96 and 108 is 864.

Note:
Let us use the second method to solve the LCM of 96 and 108.
The second method is the LCM formula calculation of the greatest common factor (GCF)
Here, the formula of LCM is as below.
$LCM\left( {a,b} \right) = \dfrac{{a \times b}}{{GCF\left( {a,b} \right)}}{\text{ }}$
The value of ‘a’ is 96, and the value of ‘b’ is 108.
Substitute the value of ‘a’ and ‘b’ in the above formula.
$ \Rightarrow LCM\left( {96,108} \right) = \dfrac{{96 \times 108}}{{GCF\left( {96,108} \right)}}{\text{ }}$...(1)
Now, find the GCF.
$ \Rightarrow 96 = {2^5} \times {3^1}$
$ \Rightarrow 108 = {2^2} \times {3^3}$
Now, let us multiply the lowest exponent prime factors to calculate the GCF of 96 and 108.
Here, the highest exponent of 2 is 5, and the highest exponent of 3 is 3.
$ \Rightarrow GCF\left( {96,108} \right) = {2^2} \times {3^1}$
Here, the value of ${2^2}$ is 4, and the value of ${3^1}$ is 3.
Therefore,
$ \Rightarrow GCF\left( {96,108} \right) = 4 \times 3$
Let us apply multiplication on the right-hand side.
That is equal to.
$ \Rightarrow GCF\left( {96,108} \right) = 12$
Put this value in equation 1.
$ \Rightarrow LCM\left( {96,108} \right) = \dfrac{{10368}}{{12}}{\text{ }}$
That is equal to,
$ \Rightarrow LCM\left( {96,108} \right) = 864$