A Complete Guide to Prime Factorization

Prime factorization, is the method of expressing the numbers as the product of its prime factors. In this method, we break down the number into its prime factors, and the product of the prime factors is the number itself.

For instance, the prime factorization of 18 is 2 × 3 × 3 or 2 × 3². 
 

A Greek mathematician named Euclid introduced the concept of prime factorization thousands of years ago. In one of his works, he stated that the presence of prime numbers is infinite. It was his discovery that set the foundation for understanding the prime numbers and factorization. Swiss mathematician Jacob Bernoulli compiled a table of prime factors up to 24,000, which served as a foundation for further discoveries. Today, prime factorization is utilized in many fields around the world. Since prime numbers are hard to break down, it is widely used in securing digital data in sectors such as banks and financial systems.

To understand the method of prime factorization, children need to learn the special characteristics of the process. There are facts about prime factorization that you might not have noticed. Let’s look at a few of its properties, given below:
 

• The fundamental theorem of arithmetic states that any number greater than 1 can be expressed as a product of its prime factors. For example, 20 = 2 × 2 × 5.
   
• When the prime factor of a number of repeats, we can express it using exponents form. For example, 16 = 2 × 2 × 2 × 2= 2⁴
   
• To find the prime factorization of the product of two numbers, we can simply combine the prime factorizations of the two numbers.
  
  For example, prime factorization of 20 and 14:
  Write the factorization of the numbers separately: 20 = 2² × 5 and 14 = 2 × 7
  Therefore, the prime factorization of 20×14 is (2² × 5) ×  (2 × 7) = 2³  × 5 × 7.
   
• When writing the prime factorization, the order of the number does not affect the result, as the multiplication is commutative. For example, 5 × 3 × 7 = 7 × 5 × 3.
   

Prime factorization is the process of breaking down the number into its prime factors. We will now learn the prime factorization through different steps:


Step 1: Verify whether the given number is divisible by 2, as it is the smallest prime number. For example, we will consider 56 as the number to factorize

56 ÷  2 =  28 (divisible by 2)

Step 2: Then check whether the number is further divisible by 2 or not. If not check divisibility using the other smallest prime numbers, such as 3, 5, 7, 11, and so on. (one by one)
 

• 28 ÷ 2 = 14 (divisible by 2)
• 14 ÷ 2 = 7 (divisible by 7)

Step 3: We will continue the process until we get the quotient as 1.
7 ÷ 7 = 1

Step 4: Note that the numbers we used to divide are the prime factors of the number.
Therefore, the prime factors of 56 are 2, 2, 2, and 7.

Step 5: If any factor repeats, write it in exponents.
The prime factorization of 56 can be written as:
56 = 23 × 7

Prime Factorization of 12

To find the prime factorization of 12
Let’s divide 12 by the smallest prime numbers: 2, 3,5, 7 

• 12 ÷ 2 =  6 (6 is divisible by 2)
• 6 ÷ 2 =  3 (3 is divisible by 3)

Check if divisible using 3

• 3 ÷ 3 =  1

Thus, the prime factorization of 12 is 2 × 2× 3 = 2² × 3 

Prime Factorization of 36

In the prime factorization of 36, we will factorize the given number by breaking it down into its prime factors. The step-by-step process is given below: 

To find the prime factorization of 36

Let’s divide 36 by the smallest prime numbers like 2, 3,5, 7 

• 36 ÷ 2 = 18 (18 is divisible by 2)
• 18 ÷ 2 = 9 (9 is divisible by 3)
   

Check if the number is divisible by 3

• 9 ÷ 3 =  3  (3 is divisible by 3)
• 3 ÷ 3 = 1

Thus, the prime factorization of 36 is 2 × 2× 3 × 3 = 2² × 3² 

 

Prime Factorization of 72

We will now find the prime factorization of 72, where the number is expressed as the product of its prime factors.

To find the prime factorization of 72

Let’s divide 72 by the smallest prime numbers like 2, 3,5, 7
 

• 72 ÷ 2 = 36 (36 is divisible by 2)
• 36 ÷ 2 = 18 (18 is divisible by 2)
• 18 ÷ 2 = 9   (9 is divisible by 3)

Check the divisibility using 3

• 9 ÷ 3 =  3  (3 is divisible by 3)
• 3 ÷ 3 = 1

Thus, the prime factorization of 72 is 2 × 2× 2× 3 × 3 = 23 × 32 
 

Prime factorization is essential for students as it is the key to understanding number theory. It can be used to find other mathematical concepts, such as HCF and LCM. To find the largest number by which they can share resources, creating a strong password, and simplifying fractions we use prime numbers. Prime factorization also helps students in preparing for math-based exams and improves their problem-solving skills. Mastering the prime factorization allows them to utilize it in securing their digital data. 
 

The prime factorization is an important concept that helps students learn other mathematical concepts. Here, we will learn how this concept can be applied as a building block in other concepts:
 

• Finding GCF and LCM
  
  We can find the largest common prime factor of the given numbers using prime factorization and LCM is about finding the smallest common multiple of the given numbers
  
  To find the GCF and LCM, we need to find the prime factorization 
  
  Let’s take 18 and 20 as the numbers,
  
  The prime factorization of 18 = 2 × 3² 
  The prime factorization of 20 = 2² × 5
  
  Hence, the GCF is 2.
  
  LCM is calculated by taking the highest powers of prime factors, that is, 2² × 3² × 5= 180.
  
   
• Simplifying Fractions
  
  We use prime factorization in converting fractions into simpler forms. It is done by canceling out the fraction (numerator and denominator) using the common prime factors.
  
  Example, Let’s simplify the fraction 20/54
  
  20 = 2² × 5
  54 = 2 × 3³
  20 / 54 =2² × 5 / 2 × 3³ 
  
  Cancelling out 2:
  2 × 5 / 3³ = 10/27 
  
  We now cancel out the common factors (2), which gives 10/27 as the result.
  
   

• Solving Algebraic Problems
  
  Prime factorization is used in simplifying the terms of algebraic problems.
  
  For example,
  
  the equation 36 p = 24 q can be solved by using prime factorization.
  
  Prime factorization of 36: 2² × 3² 
  Prime factorization of 24: 2³ × 3
  
  Simplifying the equation by dividing both sides by 36 gives: x/y = 2/3.
  
  Apply the prime factors in the given equation 
  (2² × 3²)p = (2³× 3)q
  
  We can simplify the expression: 3p = q
   

We now understand that learning prime factorization helps students in different ways. But students find it difficult to memorize the methods, so here are a few tips and tricks that help you master prime factorization:
 

• A basic understanding of prime numbers and how to factorize them helps in grasping the method.
   
• For easier calculation, the students should learn the divisibility rules of the smallest prime numbers.
   
• Using a factor tree helps children visualize the process of prime factorization
   
• Use the division ladder to find the prime factors of the given numbers, where you divide the number continuously using the smallest prime numbers until you reach 1
   

Today, the world is technologically advanced, and we understand how the prime factors are used. The prime factorization is used in several real-world situations. Children may not realize how often prime factors are used in everyday life . It is used in cryptography, coding theory, and solving mathematical problems. The prime factors are essential for securing digital data.