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What is the Prime Factorisation Method for HCF?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Prime Factorisation Method for HCF is a way to find the Highest Common Factor (HCF) of two or more numbers by breaking them down into their prime factors. HCF is the largest number that divides two or more numbers exactly, without leaving any remainder.
Simple Example
Quick Example
Imagine you have two cricket teams. Team A scored 24 runs and Team B scored 36 runs. To find the largest number of runs they could have scored in common for a certain type of boundary (like all fours or all sixes), you'd use HCF. The prime factorisation method helps us find this common factor easily.
Worked Example
Step-by-Step
Let's find the HCF of 12 and 18 using the Prime Factorisation Method.
Step 1: Find the prime factors of 12.
12 = 2 x 6
12 = 2 x 2 x 3
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Step 2: Find the prime factors of 18.
18 = 2 x 9
18 = 2 x 3 x 3
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Step 3: List the prime factors for both numbers.
Prime factors of 12: 2, 2, 3
Prime factors of 18: 2, 3, 3
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Step 4: Identify the common prime factors. These are the factors that appear in BOTH lists.
Common prime factors: 2, 3
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Step 5: Multiply the common prime factors to get the HCF.
HCF = 2 x 3 = 6
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So, the HCF of 12 and 18 is 6.
Why It Matters
Understanding HCF is super important for many real-world problems, from sharing things equally to complex computer programs. Engineers use it to design circuits, and even data scientists use similar logic to organise information efficiently. It's a foundational skill for problem-solving in science and technology.
Common Mistakes
MISTAKE: Not breaking numbers down into ONLY prime factors (e.g., writing 12 = 4 x 3). | CORRECTION: Always ensure all factors are prime numbers (2, 3, 5, 7, 11, etc.). Keep dividing until you only have prime numbers.
MISTAKE: Multiplying ALL prime factors of both numbers instead of only the COMMON ones. | CORRECTION: After finding prime factors for each number, only select the factors that appear in *every* number's list of prime factors. Then multiply *only* those common factors.
MISTAKE: Confusing HCF with LCM (Least Common Multiple). | CORRECTION: HCF means the *largest* factor common to the numbers. LCM is the *smallest* multiple that is common to the numbers. They are different concepts.
Practice Questions
Try It Yourself
QUESTION: Find the HCF of 10 and 15 using the Prime Factorisation Method. | ANSWER: HCF = 5
QUESTION: What is the HCF of 28 and 42? | ANSWER: HCF = 14
QUESTION: A shopkeeper has 30 samosas and 45 jalebis. She wants to pack them into identical boxes, with each box containing the same number of samosas and the same number of jalebis, without mixing them. What is the maximum number of boxes she can make? (Hint: Find the HCF of 30 and 45) | ANSWER: HCF = 15. So, she can make a maximum of 15 boxes.
MCQ
Quick Quiz
Which of the following is the HCF of 20 and 30?
2
5
10
60
The Correct Answer Is:
C
Prime factors of 20 are 2 x 2 x 5. Prime factors of 30 are 2 x 3 x 5. The common prime factors are 2 and 5. Multiplying them gives 2 x 5 = 10.
Real World Connection
In the Real World
When tailors in India cut fabric for many shirts, they often need to find the largest common length of fabric they can cut from different rolls to minimise waste. This is exactly what HCF helps them figure out. Similarly, when arranging rows of plants in a garden or setting up chairs for an event, HCF helps ensure equal spacing and maximum efficiency.
Key Vocabulary
Key Terms
PRIME NUMBER: A whole number greater than 1 that has only two factors: 1 and itself. E.g., 2, 3, 5, 7 | FACTOR: A number that divides another number exactly, without leaving a remainder. E.g., 2 is a factor of 4 | PRIME FACTORISATION: Breaking down a number into a product of its prime numbers. E.g., 12 = 2 x 2 x 3 | HCF (Highest Common Factor): The largest number that divides two or more numbers exactly.
What's Next
What to Learn Next
Great job learning HCF! Next, you should explore the 'Least Common Multiple (LCM)' using the Prime Factorisation Method. Understanding LCM builds on your HCF knowledge and is another vital concept in mathematics for solving more complex problems.
