Prime Factorisation

Prime factorisation is an important concept in number theory. It is used in cryptography, finding GCF and LCM, simplifying fractions, and numerous other areas. We use it often in everyday life without realizing it. It breaks down a number using prime factors. This guide will help you understand the concepts in great detail.

1.0What is Prime Factorisation?

Prime factorisation is a method to break down a number using only prime numbers. The original number is evenly divisible by these numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.

For example:

• The prime factorisation of 12 is 2 × 2 × 3.
• The prime factorisation of 30 is 2 × 3 × 5.

Each number is broken down into prime factors of a number until you can’t break it any further using only prime numbers.

2.0Prime Factorisation Method

There are three common prime factorisation methods used to find the prime factors of a number.

1. Division Method

This involves dividing the number by the smallest prime number possible (starting with 2) until the quotient becomes 1.

Steps:

• Divide the number by the smallest prime number.
• Continue dividing the quotient by prime numbers.
• Stop when the final quotient is 1.

For example, the prime factorisation of 225 using the division method can be done as

2. Factor Tree Method

Also known as the prime factor tree, this method involves splitting the number into factor pairs until all branches end in prime numbers.

Steps:

• Write the number at the top.
• Break it into two factors.
• Continue factoring non-prime numbers.
• The leaves at the end of the tree are the prime factors.

For example, the prime factors of 225, using the factor tree method, can be calculated as: 

3.0Prime Factorisation Examples

To understand the methods clearly, here are a few prime factorisation examples:

• Example 1: Prime Factorisation of 60

Using the Division Method:

60 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

Prime Factorisation of 60 = 2 × 2 × 3 × 5

Using a Prime Factor Tree:

End leaves: 2, 3, 2, 5

Here also, the Prime Factorisation of 60 is 2 × 2 × 3 × 5

• Example 2: Prime Factorisation of 100

Using the Division Method:

100 ÷ 2 = 50
50 ÷ 2 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1

Prime Factorisation of 100 = 2 × 2 × 5 × 5

• Example 3: Prime Factorisation of 84

Using Prime Factor Tree:

End leaves: 2, 2, 3, 7

Prime Factorisation of 84 = 2 × 2 × 3 × 7

4.0Prime Factors of a Number

The prime factors of a number are the set of prime numbers that multiply together to form the original number. These prime factors are unique, which is known as the Fundamental Theorem of Arithmetic. Every integer greater than 1(one) is either a prime number or may be represented as a unique product of specific prime numbers, up to the order of the factors.

5.0Prime Factorisation of Numbers from 1 to 20

Here’s a table showing the prime factorisation of numbers from 1 to 20.

6.0Importance of Prime Factorisation

Understanding the importance of prime factorisation goes beyond solving basic math problems. It has many applications in mathematics and beyond.

• Simplifying Fractions: Prime factorisation makes it easier to simplify fractions by cancelling out common prime factors in the numerator and denominator.
• Finding Least Common Multiple (LCM) and Highest Common Factor (HCF): By breaking numbers into their prime components, it's easy to calculate the LCM and HCF, which are essential for solving word problems and algebra.
• Cryptography: Modern encryption algorithms, like RSA, rely on the difficulty of factoring large numbers into their prime components, making prime factorisation vital for data security.
• Solving Algebraic Expressions: When factoring algebraic expressions, knowledge of prime numbers can help in decomposing expressions for simplification.
• Understanding Number Properties: Studying the prime factors of a number helps analyse whether a number is divisible by another, whether it’s a square or cube, and helps in determining its properties like parity or primality.

7.0Real-Life Applications of Prime Factorisation

1. Computer Science & Cryptography – Encryption algorithms use large prime numbers as a backbone of digital security systems.
2. Engineering – Prime factorisation is used in signal processing and control system analysis.
3. Banking & Security – Securing passwords and sensitive data transmission relies heavily on the difficulty of reverse prime factorisation.
4. Mathematics Education – It forms a basis for understanding division, fractions, and number systems.