Prime Factorization

Prime factorization is the method of expressing a number 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 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 a prime factor of a number repeats, we can express it using the exponent 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 a number into its prime factors. We will now learn prime factorization step-by-step:
 

Step 1: Verify whether the given number is divisible by 2, as it is the smallest prime number. To understand this with an example, let's factorize the number 56.

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 other smallest prime numbers, such as 3, 5, 7, 11, etc., 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:  The numbers 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 3 is divisible by 3
 

• 3 ÷ 3 = 1

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

Prime Factorization of 36

For 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. Prime factorization helps in finding the greatest number for equal sharing of resources, in creating strong passwords, and in simplifying fractions. Prime factorization also helps students in preparing for math-based exams and improves their problem-solving skills. 

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

 

• Finding GCF and LCM
  
  We can find the largest common prime factor of the given numbers using prime factorization. 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 to convert 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)
  
  Canceling 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 36p = 24q 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: p/q = 2/3.
  
  Substitute the prime factors in the given equation 
  
  (2² × 3²)p = (2³ × 3)q
  
  We can simplify the expression as 3p = q

We now understand that learning prime factorization helps students in different ways. Students often find it difficult to memorize the methods. Here are a few tips and tricks to 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.
   
• Start with the smallest prime. Always begin by dividing with 2 before moving to 3, 5, 7, etc. This keeps the process systematic.

Prime factorization has various real world applications. In this section, we will learn more about them.

Cryptography (Cybersecurity): Prime factorization is the backbone of RSA encryption, used in online banking, emails, and secure communication.

Computer algorithms: They are used in data compression, coding theory, and error detection/correction in digital systems.

Music rhythms: Beats are sometimes broken into factors to create rhythm cycles.

Engineering and architecture: Helps in designing equal partitions (e.g., tiling a floor of 120 tiles evenly).

Scheduling and time management: Prime factorization helps in dividing time into equal slots.