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What is the Highest Common Factor (HCF)?
Grade Level:
Class 5
Maths, Computing, Algorithms, Number Theory
Definition
What is it?
The Highest Common Factor (HCF) is the largest number that can divide two or more given numbers exactly, without leaving any remainder. It's like finding the biggest shared 'building block' for those numbers.
Simple Example
Quick Example
Imagine you have two cricket teams. Team A scored 12 runs and Team B scored 18 runs. The HCF of 12 and 18 is 6. This means 6 is the largest number that can divide both 12 and 18 perfectly.
Worked Example
Step-by-Step
Let's find the HCF of 20 and 30 using the Prime Factorisation Method.
Step 1: Find the prime factors of 20.
20 = 2 x 10
20 = 2 x 2 x 5
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Step 2: Find the prime factors of 30.
30 = 2 x 15
30 = 2 x 3 x 5
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Step 3: Identify the common prime factors. Both 20 and 30 share one '2' and one '5'.
Common factors = 2 and 5
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Step 4: Multiply the common prime factors to get the HCF.
HCF = 2 x 5 = 10
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Answer: The HCF of 20 and 30 is 10.
Why It Matters
HCF helps us simplify fractions, solve problems in geometry, and even divide things equally. Engineers use HCF to design structures efficiently, and computer programmers use it in algorithms for data processing. Learning HCF is a step towards thinking logically and solving complex problems!
Common Mistakes
MISTAKE: Finding all common factors but not picking the highest one. | CORRECTION: After listing all common factors, always select the largest number from that list.
MISTAKE: Confusing HCF with LCM (Least Common Multiple). | CORRECTION: HCF is the *largest* common divisor, while LCM is the *smallest* common multiple.
MISTAKE: Missing some prime factors when doing prime factorisation. | CORRECTION: Always divide by prime numbers (2, 3, 5, 7, 11...) until you can't divide any further, and double-check your list.
Practice Questions
Try It Yourself
QUESTION: What is the HCF of 15 and 25? | ANSWER: 5
QUESTION: Find the HCF of 18, 27, and 36. | ANSWER: 9
QUESTION: Two ropes are 42 meters and 56 meters long. What is the greatest possible length of pieces that can be cut from both ropes without any leftover? | ANSWER: 14 meters
MCQ
Quick Quiz
Which of these is the HCF of 12 and 20?
2
4
6
10
The Correct Answer Is:
B
The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 20 are 1, 2, 4, 5, 10, 20. The common factors are 1, 2, 4. The highest among these is 4.
Real World Connection
In the Real World
Imagine you are a textile shop owner in Surat. You have two rolls of fabric, one 75 meters long and another 100 meters long. To cut both rolls into pieces of equal maximum length without any waste, you'd use HCF. The HCF of 75 and 100 is 25, meaning you can cut both rolls into 25-meter pieces.
Key Vocabulary
Key Terms
FACTOR: A number that divides another number exactly | COMMON FACTOR: A factor shared by two or more numbers | PRIME NUMBER: A number greater than 1 that has only two factors: 1 and itself | PRIME FACTORISATION: Breaking down a number into its prime factors
What's Next
What to Learn Next
Great job understanding HCF! Next, you should explore the Least Common Multiple (LCM). It's another important concept that uses factors and multiples, and it will help you solve even more exciting math problems, like adding fractions with different denominators!
