Applications of Prime Factorization

Prime factorization is finding the prime numbers that make up the original number. 

Let’s take the example of 32. We know that 32 is 8 × 4, but 8 and 4 are not prime numbers. We also know that 8 is 2 × 2 × 2 and 4 is 2 × 2. Since 2 is a prime number, the prime factorization of 32 can be 2 × 2 × 2 × 2 × 2

Now that you have understood these basics, let us understand prime factorization better.

In this section, we will explore prime factorization as well as the fundamental law of arithmetic using real-life examples.

Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting practice questions on applications of prime factorization at the end of the page. 

Lesson Plan

Prime factorization of any number means to represent that number as a product of prime numbers. 

For example, prime factorization of 36 is the representation of 36 as a product of prime numbers and can be done in the following way-

Here  2 and 3 are prime numbers.

Let us see the prime factorization of a few more numbers in the following table:

Numbers	Prime Factorization
18	\( 2 \times 3^2\)
24	\( 2^3 \times 3\)
36	\( 2^2 \times 3^2\)
60	\( 2^2 \times 3 \times 5 \)
72	\( 2^3 \times 3^2\)
45	\( 3^2 \times 5 \)
40	\( 2^3 \times 5 \)
50	\( 2 \times 5^2\)
48	\( 2^4 \times 3\)
30	\( 2 \times 3 \times 5\)
42	\( 2 \times 3 \times 7\)

To find the HCF and LCM of two numbers, we use the fundamental theorem of arithmetic.

For this, we first find the prime factorization of both the numbers. Next, we consider the following:

• HCF is the product of the smallest power of each common prime factor.
• LCM is the product of the greatest power of each common prime factor.

Example:

What is the HCF of \(850\) and \(680\)?

Solution:

We first find the prime factorizations of these numbers.

Thus:

\[ \begin{align}
850&=2^{1} \times 5^{2} \times 17^{1}\\[0.3cm]680&=2^{3} \times 5^{1} \times 17^{1}
\end{align} \]

HCF is the product of the smallest power of each common prime factor.

Hence,

\[\text{HCF }(850, 680) = 2^1 \times 5^1 \times 17^1 = 170\]

LCM is the product of the greatest power of each common prime factor.

Hence,

\[\text{LCM }(850, 680) = 2^3 \times 5^2 \times 17^1 = 3400\]

Thus,

\[\text{HCF }(850, 680) =  170\\[0.3cm]\text{LCM }(850, 680) =  3400\]

Example 1

 

 

Can you help John to express \(1848\) as the product of prime factors?  Also, can you tell if this factorization is unique?

Solution

We will find the prime factorization of \(1848\):

Thus,

\(1848=2^{3} \times 3 \times 7\times 11\)

The above prime factorization is unique by the fundamental theorem of arithmetic.

Example 2

 

 

Sam wants to find the HCF of \(126, 162\) and \(180\). Can you help her find the answer to this problem?

Solution

We will find the prime factorizations of \(126, 162\) and \(180\)

Thus,

\[\begin{align}126&=2^{1} \times 3^{2} \times 7^{1}\\[0.3cm]162&=2^{1} \times 3^{4}\\[0.3cm]180&=2^{2} \times 3^{2} \times 5^{1}\end{align}\]

The HCF is the product of the smallest power of each common prime factor.

Hence,

\[\text{HCF }(126,162,180) = 2^1 \times 3^2 = 18\]

\[\text{HCF }(126,162,180)  = 18\]

Example 3

 

 

Patricia was given a task by her teacher to find the LCM of \(48\) and \(72\) using the prime factorization. Can you help her?

Solution

We will find the prime factorizations of \(48\) and \(72\)

The LCM is the product of the greatest power of each common prime factor.

Hence,

\[\text{LCM }(48, 72) = 2^4 \times 3^2 = 144\]

Thus,

\[\text{LCM }(48, 72)  = 144\]

Example 4

 

 

Jennifer has to prove that the prime factorization of 40 will always remain the same. Can you help her prove it?

Solution

Jennifer can use the division method and factor tree method to prove that the prime factorization of 40 will always remain the same.

Jennifer knows that 40 can be factored in as 5 and 8

8 can further be broken down as a product of 2 and 4

5 is a prime number already. 

Hence, she will show the division method and factor tree method in the following way:

\(\therefore\) This shows that by any method of factorization, the prime factorization remains the same.

Prime factorization for a number is unique.

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

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