Prime Factorisation
1.0Master Prime Factorisation in Minutes
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.
2.0Learning Outcomes
After completing this topic, you will be able to:
- Define prime numbers and composite numbers.
- Express any composite number as a product of prime numbers.
- Apply the Fundamental Theorem of Arithmetic.
- Find the HCF and LCM using prime factorisation.
- Solve divisibility and factorisation problems efficiently.
- Determine whether the decimal expansion of a rational number terminates or repeats using prime factors.
- Solve NCERT, competency-based, and CBSE Board examination questions confidently.
3.0What 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.
4.0Prime Factorisation Method
There are three common prime factorisation methods used to find the prime factors of a number.
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
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:
5.0Prime 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
6.0Prime 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.
7.0Prime Factorisation of Numbers from 1 to 20
Here’s a table showing the prime factorisation of numbers from 1 to 20.
8.0Importance 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.
9.0Real-Life Applications of Prime Factorisation
- Computer Science & Cryptography – Encryption algorithms use large prime numbers as a backbone of digital security systems.
- Engineering – Prime factorisation is used in signal processing and control system analysis.
- Banking & Security – Securing passwords and sensitive data transmission relies heavily on the difficulty of reverse prime factorisation.
- Mathematics Education – It forms a basis for understanding division, fractions, and number systems.
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11.0Supporting Study Materials
This study material, including CBSE Notes and NCERT Solutions for Chapter 1 of Class 10 Mathematics, is structured according to the latest NCERT guidelines. Complete with step-by-step numerical division tracks, remainder boundary highlights, and high-yield subjective proofs, this guide ensures absolute structural confidence for your school assessments and board examinations.
12.030-Second Quick Revision: Prime Factorisation
Here are a few important points to remember about Euclid's division lemma.
- Prime numbers have exactly two factors.
- Composite numbers have more than two factors.
- Every composite number has a unique prime factorisation.
- Fundamental Theorem of Arithmetic forms the basis of factorisation.
- HCF = Product of common prime factors with the smallest powers.
- LCM = Product of all prime factors with the greatest powers.
- A rational number has a terminating decimal expansion only if the denominator contains prime factors 2 and/or 5 after simplification.
- Prime factorisation is widely used in divisibility, HCF, LCM, and number theory.
13.0Previous Year Questions (PYQs) on Prime Factorisation
Question: Find the HCF of 72 and 120 using prime factorisation.
Answer
72 = 2³ × 3²
120 = 2³ × 3 × 5
Common prime factors:
2³ × 3, HCF = 24
Question: A school purchases 48 notebooks and 72 pens to distribute equally among students without any item being left over. What is the maximum number of students who can receive the materials equally?
Answer: Find the HCF of 48 and 72.
48 = 2⁴ × 3
72 = 2³ × 3²
HCF = 2³ × 3
= 24
Therefore, 24 students can receive the materials equally.
14.0Recommended Next Topics