Prime Factorization: Properties, Methods, and Examples

Prime factorization is the process of writing a number as a product of prime numbers. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves. When we break down a number using prime factorization, we are expressing it using only prime numbers.

This concept is useful in many areas of mathematics, such as finding the highest common factor (HCF), the lowest common multiple (LCM), simplifying fractions, and solving equations. It helps students understand the structure of numbers and how they are formed.

In this blog, you will learn different methods to find the prime factorization of a given number with solved examples to make the concept easier to understand.

What is a Prime Number?

To understand the concept of prime factorization, let us first understand the meaning of prime numbers 

A prime number is a natural number greater than 1 that has exactly two distinct factors i.e., 1 and the number itself. This means it can only be divided evenly (without a remainder) by 1 and by itself.  A few examples of prime numbers include:

2, 3, 5, 7, 11, 13, 17, 19:

The number 2 is the smallest prime number and also the only even prime. All other even numbers are divisible by 2 and hence have more than two factors, which disqualifies them from being prime.

What is a Prime Factor?

Now that we know what a prime number is, we can define a prime factor.

A prime factor of a number is a prime number that divides the given number exactly, meaning it leaves no remainder. When a number is expressed as a product of prime numbers, those primes are called its prime factors.

For example, consider the number 30:

• 30 can be written as 2 × 3 × 5.

• Each of these (2, 3, and 5) is a prime number.

• Therefore, 2, 3, and 5 are the prime factors of 30.

It is important to note that while a number may have many factors, not all of them are prime. Prime factors are specifically those that are both divisors and prime.

Read More - Co Prime Numbers 

What is Prime Factorization?

Now that we have understood the meaning of prime numbers and prime factors, let’s now understand what prime factorization means.

Prime factorization is the process of breaking down a composite number into a product of prime numbers. In simpler terms, it is the method of expressing a number as the multiplication of its prime factors.

Every natural number greater than 1 is either a prime number or can be uniquely written as a product of prime numbers. This concept is known as the Fundamental Theorem of Arithmetic, which states that the prime factorization of a number is unique, except for the order of the factors.

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Properties of Prime Factorization

Prime factorization has several important properties. Below, we discuss a few of the most commonly used ones:

Applies to all whole numbers above 1:  Every number greater than 1 can be broken down into a product of prime numbers.

Always gives a unique result: The prime factorization of a number is always the same, regardless of the method used.

Useful for finding HCF and LCM: Comparing prime factors makes it easier to calculate the highest common factor and the least common multiple.

Only valid for natural numbers: Prime factorization does not apply to decimals, fractions, or negative values.

Simplifies fractions easily: Common prime factors in the numerator and denominator can be canceled out to simplify fractions.

Used in data encryption: Cryptography algorithms often rely on the difficulty of factorizing large numbers into primes.

Can be shown using factor trees: A factor tree helps visually break down a number into its prime components step by step.

Read More -What are Whole Numbers? Definition and Examples

Methods to Find Prime Factorization

There are two common methods used to find the prime factorization of a number.

•  The Division Method and 

• The Factor Tree Method. 

 Let’s understand each method in detail:

The Division Method

In this method, we repeatedly divide the number by the smallest prime numbers until the final result is 1.

Steps to Find  Prime Factorization Using the Division Method

Step 1: Begin by dividing the number by the smallest prime number, which is 2. Continue dividing by 2 until it no longer divides evenly.

Step 2: Once division by 2 is no longer possible, move to the next smallest prime (3, then 5, then 7, and so on), dividing at each step.

Step 3: Continue the process until the quotient becomes 1.

Step 4: The prime numbers used in each step of division are the prime factors of the original number.

Let’s understand the above steps with an example below: 

Example 1: Prime Factorization of 144 using the division method

144 ÷ 2 = 72

72 ÷ 2 = 36

36 ÷ 2 = 18

18 ÷ 2 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

The prime factorization of 144 is 2 × 2 × 2 × 2 × 3 × 3, or 2⁴ × 3².

Read More - Even Numbers: Definition, Example, Properties, List 1 to 1000

Example 2: Prime Factorization of 180 using the division method

180 ÷ 2 = 90

90 ÷ 2 = 45

45 ÷ 3 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

The Prime factorization of 180 is 2 × 2 × 3 × 3 × 5, or 2² × 3² × 5

Steps to Find Prime Factorization Using the Factor Tree Method

Follow these steps to find the prime factors of a number using the factor tree:

Step 1: Start with the given number.

Step 2: Choose any two factors of the number (they don’t need to be prime).

Step 3: If a factor is not prime, break it down further into two factors.

Step 4: Repeat the process until all resulting numbers are prime.

Step 5: The prime numbers at the ends of the branches are the prime factors. Multiply them to verify the original number.

Read More - Ordinal Numbers 1 to 100: How to Write, Applications, Fun Facts

Example 1: Prime Factorization of 60 Using the Factor Tree Method

Step 1: Start with the number 60

Step 2: Break it into any two factors
60 = 6 × 10

Step 3: Now, break down each factor further

6 = 2 × 3 (both prime)

10 = 2 × 5 (both prime)

Step 4: Now we have got all prime numbers i.e, 2, 2, 3, and 5.

So, the prime factorization of 60 is 2 × 2 × 3 × 5, or 2² × 3 × 5

Example 2: Prime Factorization of 72 Using the Factor Tree Method.

Step 1: Start with the number 72.

Choose two factors of 72 

72 = 8 × 9

Step 2: Now factor each of those:

8 = 2 × 4
4 = 2 × 2
So, 8 = 2 × 2 × 2

9 = 3 × 3

Step 3: All resulting numbers are prime 

2, 2, 2, 3, and 3
The  Prime factorization of 72 = 2 × 2 × 2 × 3 × 3 or written with exponents: 2³ × 3².

Read More - Integers: Types, Rules, Properties, and Examples

HCF and LCM Using Prime Factorization

The method of prime factorization can be effectively used to find both the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two or more numbers. Let’s understand the steps below:

Steps to Find the HCF Using Prime Factorization

To find the HCF using prime factorization, follow the steps below:

Step 1: Write the prime factorization of each number.
Step 2: Identify the common prime factors between the numbers.

Step 3: For each common factor, choose the lowest exponent (power).

Step 4: Multiply these lowest powers together to get the HCF.

Example: Find the HCF of 48 and 60 using Prime Factorization

Step 1: Find the Prime Factorization of 48 nd 60

48 = 2⁴ × 3

60 = 2² × 3 × 5

Step 2: Identify the Common Prime Factors

The common prime factors between 48 and 60 are 2 and 3

Now compare the powers of each common factor:

For 2, the lowest power is 2²

For 3, the lowest power is 3¹

Step 3: Multiply the Lowest Powers of the Common Factors

HCF = 2² × 3 = 4 × 3 = 12

Steps to Find the LCM Using Prime Factorization

To find the LCM using prime factorization, follow the steps below:

Step 1: Write the prime factorization of each number.

Step 2: List all prime factors that appear in either number.

Step 3: For each prime, choose the highest exponent (power).

Step 4: Multiply these highest powers together to get the LCM.

Read More - Real Numbers: Meaning, Symbol, Properties, and Solved Examples

Example: Find the LCM of 48 and 60 Using Prime Factorization

Step 1: Find the Prime Factorization of  48 and 6

• 48 = 2⁴ × 3

• 60 = 2² × 3 × 5

Step 2: List All Prime Factors Involved

The prime factors that appear in either number are 2, 3, and 5

Step 3: Take the Highest Power of Each Prime Factor

For 2, the highest power is 2⁴

For 3, the highest power is 3¹

For 5, the highest power is 5¹

Step 3: Multiply the Highest Powers Together

LCM = 2⁴ × 3 × 5
  = 16 × 3 × 5
  = 48 × 5
  = 240 

LCM of 48 and 60 is 240.

Prime factorization is a way to break down numbers using only prime numbers. It makes solving maths problems like HCF, LCM, and simplification much easier. With regular practice, students can strengthen their understanding of this concept. Help your child learn to solve maths problems like prime factorization and many other important topics using mental math tricks by joining CuriousJr maths online classes. Our friendly teachers make learning simple, clear, and enjoyable for every curious young mind. Book a demo today.