Prime Factorization

Prime factorization is the process of breaking down a number. It breaks down a number to write it in the form of its prime factors. In this method, we write the number in a way that the product of the prime factors is the number itself. We can perform prime factorization using methods such as the division method or the factor tree method.

Let us now take a look at how prime factorization works.

Prime factorization of 24 is given as: \(24=2\times2\times2\times3=2^3\times3\)

Prime factorization of 36 is given as: \(36=2\times2\times3\times3=2^2\times3^2\)

Prime factorization of 72 is given as: \(72 = 2\times2\times2\times3\times3=2^3\times3^2\)

Prime factorization of 48 is given as: \(48 = 2\times2\times2\times2\times3=2^4\times3\)

Prime factorization of 54 is given as: \(54=2\times3\times3\times3=2\times3^3\)

Prime factorization of 52 is given as: \(52=2\times2\times13=2^2\times13\)

Prime factorization of 60 is given as: \(60=2\times2\times3\times5=2^2\times3\times5\)

Prime factorization of 45 is given as: \(45= 3\times3\times5=3^2\times5\)

Prime factorization of 32 is given as: \(32=2\times2\times2\times2\times2=2^5\)

Prime factorization of 75 is given as: \(75=5\times5\times3=5^2\times3\)

• Euclid, an ancient Greek mathematician, established that prime numbers are infinite, laying the groundwork for prime factorization.
   
• Later, Swiss mathematician Jacob Bernoulli created a table of prime factors up to 24,000, aiding further research.
   
• Today, prime factorization is widely used, especially in securing digital data in banking and finance, due to the complexity of prime numbers.

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, \(16 = 2 × 2 × 2 × 2 = 2^4\)
   
• 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: \(20 = 2^2 × 5\) and \(14 = 2 × 7\)
  
  Therefore, the prime factorization of \(20 × 14\) is \((2^2 × 5) × (2 × 7) = 2^3 × 5 × 7\).
   
• When writing the prime factorization, the order of the number does not affect the result, as the multiplication is commutative. For example, \(5 × 3 × 7 = 7 × 5 × 3\).

The prime factorization can be done on any number in two different methods. They are, division method and factor tree method. We will now learn the division method.

Division method: We can find the factors of a number by dividing the given number with only prime numbers. Let us learn this method with the help of an example.

Example: Find the prime factors of 72 using the division method.

Step 1: Firstly, we have to divide 72 with its smallest prime factor, which is 2. Therefore, we get \(\frac{72}{2}=36\).

Step 2: Repeat the process until we get a number that is not any more divisible by 2. \(\frac{36}{2}=18\)

Step 3: Let us divide 18 by 2. \(\frac{18}{2}=9\)

Step 4: Since we cannot divide 9 by 2, let us divide the number with the smallest prime factor of 9. Therefore, we get \(\frac{9}{3}=3\).

Step 5: Divide the number till we get 1 as the quotient. \(\frac{3}{3}=1\)

Step 6: Now we got the prime factors of 72. It can be written as, \(72 = 2\times2\times2\times3\times3=2^3\times3^2\). Where, 2 and 3 are prime numbers and the prime factors of 72.

Factor tree method: We can also find the prime factors of any given number with the help of factor tree method. In this process, we keep on factorizing the given number until we find its prime factors. The factors are split and written in the form of branches of a tree. We circle the final factors and consider them as the prime factors of the given number. Let us understand this process by finding the prime factors of 72.

Step 1: Firstly, split 72 into two factors. Let us take 18 and 4.

Step 2: Check if these two numbers are prime or not. 

Step 3: Since the two numbers are composite, they can be divided further into more factors. Repeat this process until we get prime numbers.

Step 4: Split 4 into 2 times 2. Similarly, 18 can be split into 2 times 9. 

Step 5: We can further split 9 into 3 times 3. Now, we are left only with prime numbers. Circle all the prime numbers at the end. 

Step 6: Therefore, the prime factors of 72 are given as, \(72= 2\times2\times2\times3\times3=2^3\times3^2\)

The factor tree is not always the same. If we split 72 into 2 and 36, the form of the tree would be different. But that doesn't change the prime factors of the given number. 

• 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. Similarly, prime factorization is the base of finding LCM and HCF. Here, we will learn how this concept can be applied in other math concepts:

We must find the primes factors of the given numbers. We can then use these prime factors to find the HCF and LCM. The HCF can be found by taking the lowest powers of the common prime numbers. Similarly, the LCM can be found by taking the highest powers of each prime number. 

For example, let’s take 18 and 20 as the numbers

The prime factorization of \(18 = 2 \times 3\times3=2\times3^2\)
 
The prime factorization of \(20 = 2\times2\times5=2^2 \times 5\)

Hence, the HCF is calculated by taking the lowest powers of the prime factors, that is, 2.

LCM is calculated by taking the highest powers of prime factors, that is, \(2^2 × 3^2 × 5 = 180\).

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.
   
• Before starting to teach prime factorization, teachers and parents should be sure that the kid understands what are factors and multiples. Teach them about the concept of prime numbers and composite numbers.
   
• Parents should encourage their children to make factor trees for a better understanding of the concept. Ask them to use different colors for the prime factors. 
   
• Teachers and parents should use some examples that are easy for the kids to understand. Use some real-life context like "You have 24 candies. How many equal groups can you make?"
   
• Teachers should them easy shortcuts and tricks like how to check if the number is divisible by 2, 3, or 5. Parents and teachers should ask them to divide the number repeatedly by the smallest factor instead of using tree method for a better speed in calculation. 

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.
   
• 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 \(36= 2^2 × 3^2\) 
  Prime factorization of \(24= 2^3 × 3\)

  
  Substitute the prime factors in the given equation 

  \((2^2 × 3^2)p = (2^3 × 3)q\)
   

  We can simplify the expression as \(3p = q\)