HCF (Highest Common Factor) Introduction
HCF (Highest Common Factor) is the largest factor that two or more numbers share. It is also called the Greatest Common Factor (GCF).
When we find all the factors of two numbers and pick the biggest one they have in common, that number is the HCF. This concept is used in simplifying fractions, dividing objects into equal groups, and solving many real-life problems.
What is HCF (Highest Common Factor) Introduction - Class 4 Maths (Factors and Multiples)?
The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides each of them exactly, leaving no remainder.
For example, the factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6. The highest among them is 6. So, HCF of 12 and 18 = 6.
HCF (Highest Common Factor) Introduction Formula
HCF = The largest number in the list of common factors
Steps to find HCF by listing factors:
- Write all factors of each number.
- Circle the factors that appear in every list.
- The greatest circled factor is the HCF.
Types and Properties
Methods to Find HCF:
- Listing Method: List all factors of each number, find common factors, pick the highest.
- Prime Factorisation Method: Write each number as a product of prime factors. Multiply the common prime factors.
Listing Method Example:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12
Prime Factorisation Method Example:
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
Common prime factors: 2 × 2 × 3 = 12
Solved Examples
Example 1: Example 1: HCF by Listing Factors
Problem: Find the HCF of 8 and 12.
Solution:
Step 1: Factors of 8 = 1, 2, 4, 8
Step 2: Factors of 12 = 1, 2, 3, 4, 6, 12
Step 3: Common factors = 1, 2, 4
Step 4: The greatest common factor = 4
Answer: HCF of 8 and 12 = 4
Example 2: Example 2: HCF by Listing Factors
Problem: Find the HCF of 15 and 25.
Solution:
Step 1: Factors of 15 = 1, 3, 5, 15
Step 2: Factors of 25 = 1, 5, 25
Step 3: Common factors = 1, 5
Step 4: The greatest common factor = 5
Answer: HCF of 15 and 25 = 5
Example 3: Example 3: HCF by Prime Factorisation
Problem: Find the HCF of 18 and 24 using prime factorisation.
Solution:
Step 1: Prime factorisation of 18 = 2 × 3 × 3
Step 2: Prime factorisation of 24 = 2 × 2 × 2 × 3
Step 3: Common prime factors = 2 × 3 = 6
Answer: HCF of 18 and 24 = 6
Example 4: Example 4: HCF of Three Numbers
Problem: Find the HCF of 12, 18, and 24.
Solution:
Step 1: Factors of 12 = 1, 2, 3, 4, 6, 12
Step 2: Factors of 18 = 1, 2, 3, 6, 9, 18
Step 3: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 4: Common factors of all three = 1, 2, 3, 6
Step 5: Greatest common factor = 6
Answer: HCF of 12, 18, and 24 = 6
Example 5: Example 5: HCF When One Number is a Factor of Another
Problem: Find the HCF of 9 and 27.
Solution:
Step 1: Factors of 9 = 1, 3, 9
Step 2: Factors of 27 = 1, 3, 9, 27
Step 3: Common factors = 1, 3, 9
Step 4: Greatest common factor = 9
Answer: HCF of 9 and 27 = 9
When one number is a factor of the other, the smaller number is the HCF.
Example 6: Example 6: HCF of Co-prime Numbers
Problem: Find the HCF of 8 and 15.
Solution:
Step 1: Factors of 8 = 1, 2, 4, 8
Step 2: Factors of 15 = 1, 3, 5, 15
Step 3: Common factor = 1
Answer: HCF of 8 and 15 = 1
Numbers whose HCF is 1 are called co-prime numbers.
Example 7: Example 7: Word Problem
Problem: Aman has 16 red marbles and 24 blue marbles. He wants to put them in groups so that each group has the same number of red marbles and the same number of blue marbles. What is the greatest number of groups he can make?
Solution:
Step 1: We need the HCF of 16 and 24.
Step 2: Factors of 16 = 1, 2, 4, 8, 16
Step 3: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 4: Common factors = 1, 2, 4, 8
Step 5: HCF = 8
Answer: Aman can make 8 groups (each with 2 red and 3 blue marbles).
Example 8: Example 8: Word Problem
Problem: Priya has 20 mangoes and 30 guavas. She wants to distribute them equally among her friends without any fruit left over. What is the greatest number of friends she can share with?
Solution:
Step 1: Find HCF of 20 and 30.
Step 2: Factors of 20 = 1, 2, 4, 5, 10, 20
Step 3: Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
Step 4: Common factors = 1, 2, 5, 10
Step 5: HCF = 10
Answer: Priya can share with 10 friends (each gets 2 mangoes and 3 guavas).
Example 9: Example 9: HCF by Prime Factorisation (Larger Numbers)
Problem: Find the HCF of 36 and 48 using prime factorisation.
Solution:
Step 1: 36 = 2 × 2 × 3 × 3
Step 2: 48 = 2 × 2 × 2 × 2 × 3
Step 3: Common prime factors = 2 × 2 × 3 = 12
Answer: HCF of 36 and 48 = 12
Real-World Applications
Where do we use HCF in daily life?
- Dividing items equally: When you want to split different items into the largest equal groups, use HCF.
- Simplifying fractions: Divide both numerator and denominator by their HCF to get the simplest form.
- Cutting materials: To cut a sheet into the largest equal square pieces, find the HCF of the length and breadth.
- Arranging things in rows: To arrange objects in the maximum number of equal rows, find the HCF.
Key Points to Remember
- The HCF of two or more numbers is the largest factor common to all of them.
- HCF is also called Greatest Common Factor (GCF) or Greatest Common Divisor (GCD).
- The HCF of two co-prime numbers is always 1.
- If one number is a factor of the other, the smaller number is the HCF.
- The HCF of any number and 1 is always 1.
- HCF is always less than or equal to the smallest of the given numbers.
- HCF is used to simplify fractions to their lowest terms.
Practice Problems
- Find the HCF of 14 and 21.
- Find the HCF of 16 and 40.
- Find the HCF of 28 and 42 using prime factorisation.
- Find the HCF of 10, 15, and 25.
- Meera has 18 pencils and 27 erasers. She wants to make gift bags with the same number of pencils and erasers in each bag. What is the greatest number of bags she can make?
- Find the HCF of 7 and 13. What kind of numbers are they?
- Rahul has a rope of 36 m and another of 48 m. He wants to cut them into pieces of equal length without any rope left over. What is the longest piece he can cut?
Frequently Asked Questions
Q1. What is HCF in maths?
HCF stands for Highest Common Factor. It is the largest number that divides two or more given numbers exactly, leaving no remainder. For example, the HCF of 12 and 18 is 6.
Q2. What is the difference between HCF and LCM?
HCF is the greatest factor common to two or more numbers, while LCM is the smallest multiple common to them. HCF is always less than or equal to the numbers, and LCM is always greater than or equal to the numbers.
Q3. How do you find HCF by listing factors?
Write all factors of each number. Then find the factors that appear in every list. The largest of these common factors is the HCF.
Q4. What are co-prime numbers?
Co-prime numbers are two numbers whose only common factor is 1, meaning their HCF is 1. For example, 8 and 15 are co-prime because HCF(8, 15) = 1.
Q5. Can HCF be greater than the given numbers?
No. The HCF is always less than or equal to the smallest of the given numbers. It can equal the smallest number only when that number is a factor of all the others.
Q6. What is the HCF of two consecutive numbers?
The HCF of any two consecutive numbers (like 7 and 8, or 14 and 15) is always 1. Consecutive numbers are always co-prime.
Q7. How is HCF used to simplify a fraction?
To simplify a fraction, divide both the numerator and denominator by their HCF. For example, to simplify 12/18, find HCF(12, 18) = 6. Then 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12/18 = 2/3.
Q8. What is the HCF of a number and itself?
The HCF of any number with itself is the number itself. For example, HCF(15, 15) = 15, because every factor of 15 is common to both.
Q9. Is HCF covered in the NCERT Class 4 syllabus?
Yes. HCF is introduced in Class 4 under the chapter Factors and Multiples. Students learn to find the HCF using the listing method and prime factorisation method.
