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What is a Factorisation?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation is like breaking down a number into smaller numbers that multiply together to give the original number. These smaller numbers are called factors. Think of it as finding the 'building blocks' of a number through multiplication.
Simple Example
Quick Example
Imagine you have 12 ladoos. You want to arrange them in equal rows. You could have 2 rows of 6 ladoos (2 x 6 = 12), or 3 rows of 4 ladoos (3 x 4 = 12). Here, 2, 6, 3, and 4 are factors of 12. Factorisation is finding these pairs of numbers.
Worked Example
Step-by-Step
Let's factorise the number 20.
---Step 1: Start with the smallest prime number, 2. Can 20 be divided by 2 without a remainder? Yes, 20 / 2 = 10.
---Step 2: Now take the result, 10. Can 10 be divided by 2? Yes, 10 / 2 = 5.
---Step 3: Now take the result, 5. Can 5 be divided by 2? No. Can it be divided by the next prime number, 3? No. Can it be divided by 5? Yes, 5 / 5 = 1.
---Step 4: We stop when we reach 1. The numbers we used to divide are 2, 2, and 5.
---Step 5: So, the prime factorisation of 20 is 2 x 2 x 5. You can check: 2 x 2 = 4, and 4 x 5 = 20.
Answer: The prime factors of 20 are 2, 2, and 5.
Why It Matters
Factorisation is super important in many advanced fields! It's used in cryptography to secure your online messages and banking transactions, making sure no one can snoop. Data scientists use it to understand complex data patterns, and engineers apply it in designing everything from bridges to computer chips.
Common Mistakes
MISTAKE: Students sometimes list all numbers that divide a number, instead of focusing on prime factors for prime factorisation. For example, for 12, listing 1, 2, 3, 4, 6, 12. | CORRECTION: For prime factorisation, only break down into prime numbers (numbers only divisible by 1 and themselves, like 2, 3, 5, 7...). So for 12, it's 2 x 2 x 3.
MISTAKE: Forgetting to continue dividing until the number becomes 1. For example, stopping at 20 = 4 x 5. | CORRECTION: Always break down composite factors (like 4) into their prime factors (4 becomes 2 x 2). So, 20 = 2 x 2 x 5.
MISTAKE: Confusing factors with multiples. For example, thinking 24 is a factor of 6. | CORRECTION: Factors divide a number evenly. Multiples are what you get when you multiply a number by another whole number. 6 is a factor of 24 (6 x 4 = 24), but 24 is a multiple of 6.
Practice Questions
Try It Yourself
QUESTION: Find the prime factorisation of 18. | ANSWER: 2 x 3 x 3
QUESTION: Which of these is NOT a prime factor of 30: 2, 3, 5, 6? | ANSWER: 6 (because 6 is not a prime number)
QUESTION: The product of two numbers is 42. If one number is a prime factor of 10, what is the other number? | ANSWER: The prime factors of 10 are 2 and 5. If one number is 2, the other is 42/2 = 21. If one number is 5, the other is 42/5 (not a whole number). So the other number is 21.
MCQ
Quick Quiz
Which of the following is the prime factorisation of 36?
2 x 18
3 x 12
2 x 2 x 3 x 3
4 x 9
The Correct Answer Is:
C
Option C (2 x 2 x 3 x 3) correctly breaks 36 down into only prime numbers. Options A, B, and D contain composite numbers (18, 12, 4, 9) which are not prime factors.
Real World Connection
In the Real World
Ever wondered how your online payments are safe? Factorisation is at the heart of it! Cryptographers use the difficulty of factorising very large numbers into their prime factors to create strong encryption. When you use UPI or NetBanking, complex factorisation helps keep your money and data secure from cyber threats.
Key Vocabulary
Key Terms
FACTOR: A number that divides another number exactly, leaving no remainder. | PRIME NUMBER: A whole number greater than 1 that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7). | COMPOSITE NUMBER: A whole number greater than 1 that has more than two factors (e.g., 4, 6, 8, 9). | PRIME FACTORISATION: Breaking down a composite number into a product of its prime factors.
What's Next
What to Learn Next
Great job understanding factorisation! Next, you should explore 'Highest Common Factor (HCF)' and 'Lowest Common Multiple (LCM)'. These concepts build directly on factorisation and are super useful for solving many real-life math problems!
