Prime Factorization

Prime Factorization is a way of writing numbers as the product of prime numbers. Prime numbers are natural numbers that have only two divisors, 1 and themselves. Prime Factorization involves only the prime numbers as every composite number can be written as the product of primes.

Let’s understand the Prime Factorization with the following illustration:

More Examples of Prime Factorization:

12 can be written as 2 x 6
6 can be further factorized as 2 x 3.
So 12 can be rewritten as 2 x 2 x 3
No more factorization possible as 2 and 3 cannot be divide further, so 2 x 2 x 3 is our prime factorization

54 can written as 2 x 27.
27 can be further factorized as 3 x 9.
So we rewrite 54 as 2 x 3 x 9
9 can be further factorized as 3 x 3.
So we rewrite 54 as 2 x 3 x 3 x 3.
No more factorization possible, so 2 x 3 x 3 x 3 is our prime factorization

In this article, we will learn about what is prime factorization, its examples, and methods in detail.

Prime Factorization Meaning

Prime factorization is the method of identifying the prime factors of a number. Since composite numbers have more than two factors, this method is applicable exclusively to them and not to prime numbers, which only have two distinct positive divisors: 1 and the number itself​.

What are Prime Factors?

Prime factors are the prime numbers that divide a given number exactly without leaving a remainder. In other words, they are the building blocks of a number.

When a number is expressed as a product of its prime factors, it is said to be in its prime factorization form.

Some examples of prime factors are:

• 2 and 3 are the prime factors of 12, as 12 = 2² × 3,
• 3 and 5 are the prime factors of 15, as 15 = 3 × 5,
• 2 and 7 are the prime factors of 14,  as 15 = 3 × 5.

Prime Factorization Methods

Two common methods of Prime Factorization are:

• Division Method
• Factor Tree Method

Prime Factorization by Division Method

In this method, the number is successively divided by prime numbers until the quotient becomes 1, with each division identifying a prime factor.

Steps to identify the prime factors of a number by Division Method :

• Step 1: Divide the number by the smallest prime number (i.e. 2) until we are able to divide the given number without leaving any remainder.
• Step 2: Move on to the next prime number and repeat the division until the quotient becomes 1.
• Step 3: The prime factors are the divisors used in the division process.

Let's consider some examples for better understanding.

Examples of Prime Factorization by Division Method

Example 1: Find the Prime Factorization of 60 using Division Method.

Example 2: Find the Prime Factorization of 210 using Division Method.

Example 3: Express 56 as the product of its Prime Factors.

Prime Factorization by Factor Tree Method

The Factor Tree Method involves breaking down a number into its prime factors by constructing a tree-like structure called a factor tree.

Steps to identify the prime factors of a number by Factor Tree Method:

• Step 1: Identify two factors of the number that are not prime.
• Step 2: Write these two factors as branches of the factor tree.
• Step 3: Repeat steps 1 and 2 for each non-prime factor until all branches end with prime numbers.
• Step 4: The prime factors are the numbers at the end of the branches.

Let's consider some examples for better understanding as follows:

Examples of Prime Factorization by Factor Tree Method

Example 1: Find the factorization of 60 by the Factor Tree Method.

Example 2: Make the Factor Tree of 210.

Please refer Prime Factorization Tips and Tricks to Improve your time in finding prime factorization.

Prime Factorization of Numbers

Some examples of prime factorization are listed below:

Finding HCF and LCM by Prime Factorization

HCF and LCM can be easily calculated by the method of prime factorization:

Finding HCF

For the HCF, take the lowest power of each common prime factor from both numbers.

For Example:

• Common prime factors: 2 and 3
• For 2: min⁡(2,4) = 2
• For 3: min⁡(1,1) = 1

So, the HCF is:

HCF = 2² x 3¹ = 4 x 3 = 12

Finding LCM

For the LCM, take the highest power of each prime factor present in either number.

For Example:

• Prime Factors: 2,3 and 5
• For 2: max⁡(2,4) = 4
• For 3: max⁡(1,1) = 1
• For 5: max⁡(1,0) = 1

So, the LCM is:

LCM = 2⁴ × 3¹ × 5¹ = 16 × 3 × 5 = 240

Applications of Prime Factorization in Real Life

• Finding HCF and LCM: Prime factorization helps determine the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers, essential for simplifying fractions and finding common denominators.
• Cryptography: It is crucial in public key cryptography, such as RSA, where the difficulty of factoring large composite numbers ensures secure communication.
• Simplifying Fractions: By factoring numerators and denominators into prime factors, common factors can be canceled out, simplifying fractions effectively.
• Divisibility Rules: Prime factorization aids in applying divisibility rules, quickly indicating whether one number is divisible by another.
• Data Compression: Techniques based on prime factorization can optimize data storage and transmission in computer science, enhancing efficiency.
• Network Security: Algorithms based on prime factorization enhance data security during network transfers, protecting sensitive information.

Read in Detail: Real Life Applications of Prime Factorization

Prime Factorization Solved Examples

Let's solve some questions on Prime Factorisation.

Problem 1: What is the Prime Factorisation of 80?

Solution:

To find the prime factorization of 80, we can start by dividing it by the smallest prime number, which is 2.

• 80 divided by 2 equals 40.
• 40 divided by 2 equals 20.
• 20 divided by 2 equals 10.
• 10 divided by 2 equals 5.

Now, since 5 is a prime number, we can stop dividing. Therefore, the prime factorization of 80 is: 2 × 2 × 2 × 2 × 5.

Problem 2: Prime factorization of 120.

Solution:

Starting with the smallest prime number, which is 2.

• 120 divided by 2 equals 60.
• 60 divided by 2 equals 30.
• 30 divided by 2 equals 15.
• Now, since 15 is not divisible by 2, we move on to the next prime number (i.e, 3)
• 15 divided by 3 equals 5.

Now, since 5 is a prime number, we can stop dividing. Therefore, the prime factorization of 120 is: 2 × 2 × 2 × 3 × 5

Problem 3: What is the Factor Tree of 56?

Related Articles:

• Interesting Facts about Prime Numbers
• Factorization of Algebraic Expression
• Composite Numbers

Prime Factorization Worksheet

1. Find the prime factorization of 36.

2. Determine the prime factorization of 90.

3. What is the prime factorization of 48?

4. Find the prime factorization of 105.

5. What is the prime factorization of 84?

6. Determine the prime factorization of 100.

7. Find the prime factorization of 2310.

8. What is the prime factorization of 56?

9. Determine the prime factorization of 150.

10. What is the prime factorization of 1250?

Answer Key

1. 36: 2^2 \times 3^2
2. 90: 2 \times 3^2 \times 5
3. 48: 2^4 \times 3
4. 105: 3 \times 5 \times 7
5. 84: 2^2 \times 3 \times 7
6. 100: 2^2 \times 5^2
7. 2310: 2 \times 3 \times 5 \times 7 \times 11
8. 56: 2^3 \times 7
9. 150: 2 \times 3 \times 5^2
10. 1250: 2 \times 5^4

Conclusion

Prime factorization is a fundamental concept in mathematics that involves expressing a number as a product of its prime factors. This technique is not only essential for simplifying mathematical expressions but also plays a crucial role in various fields, including number theory, cryptography, and computational mathematics. In summary, prime factorization is a vital mathematical tool that underpins many concepts and applications. Its significance extends beyond pure mathematics into practical applications in technology and everyday life.