HCF

Introduction to HCF

HCF stands for the highest common factor. This is the largest number that can exactly divide two or more numbers. Sometimes it is also called GCD (greatest common divisor). For example, the HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

We can find the HCF of numbers in different ways. Some common methods are listing factors, prime factorization, and the division method. These methods help us easily find the HCF of any set of numbers, whether small or large.

HCF is very useful in everyday life. It helps to divide things into equal parts, simplify fractions, and solve problems in math. Learning from HCF steps with examples makes it easy to understand and use in different situations.

Table of Contents:

• Definition
• Method to Calculate
• By Prime Factorization
• By Division Method
• HCF of 3 Numbers
• HCF of 4 Numbers
• HCF of Prime Numbers
• Properties
• Solved Examples
• Word Problems
• Practice Problems
• FAQs on HCF

 

Definition

HCF stands for Highest Common Factor. It is the largest number that divides two or more given numbers exactly (without leaving any remainder). In simple words, HCF is the biggest number that is a factor of all the given numbers. Factors are numbers that multiply together to give another number.

Example: Find the HCF of 24 and 36.

• 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
  
  

Method to Calculate

We can find the HCF of any given numbers by using two methods:

• By the prime factorization method 

• By the division method

Let us discuss these two methods one by one in this article.

 

By Prime Factorization

• Write the prime factors of each number.

• Identify the common prime factors.

• Multiply them to get the HCF

Example 1: Find the HCF of 30 and 45

• 30 = 2 × 3 × 5

• 45 = 3 × 3 × 5

• Common prime factors = 3 × 5 = 15

Example 2: Find the HCF of 60 and 75

• Prime factors of 60 = 2² × 3 × 5

• Prime factor of 75 = 3 × 5²

• Common prime factor = 3 and 5

• Lowest power of 3 = 3, lowest power 5 = 5 

• HCF = 3 × 5 = 15

Example 3: Find the HCF of 36, 27, and 80

• Prime factors of 36 = 2² × 3²

• Prime factors of 27 = 3³

• Prime factors of 80 = 2⁴ × 5

• Common prime factors = none

• HCF = 1

 

By Division Method

We can also find the highest common factor (HCF) using the division method. This method uses repeated division to find common factors.

Steps to find HCF by the division method:

1. Write the given numbers side by side, separated by commas.

2. Divide them by the smallest prime number that can divide them all exactly.

3. Write quotients under each number.

4. Repeat the process until there are no more common prime numbers left.

5. Multiply all the prime numbers that divide the numbers exactly. This product is HCF.

Example 1: Find the HCF of 30 and 75

• 30 and 75 are both divisible by 3 → Quotients are 10 and 25.

• 10 and 25 are both divisible by 5 → Quotients are 2 and 5.

• No more common factors are left. HCF = 3 × 5 = 15

Example 2: Find the HCF of 36 and 24

• 36 and 24 are both divisible by 2 → Quotients are 18 and 12.

• 18 and 12 are both divisible by 2 → Quotients are 9 and 6.

• 9 and 6 are both divisible by 3 → Quotients are 3 and 2.

• No more common factors are left. HCF = 2 × 2 × 3 = 12

Example 3: Find the HCF of 36, 12, 24, and 48

• All are divisible by 2 → Quotients are 18, 6, 12, and 24.

• Again divisible by 2 → Quotients are 9, 3, 6, and 12.

• Again divisible by 3 → Quotients are 3, 1, 2, 4.

• No more common factors are left. HCF = 2 × 2 × 3 = 12
  
  

HCF of 3 Numbers

To find the HCF of three numbers, we find the number that divides all three exactly.

• First, find the HCF of any two numbers.

• Then, find the HCF of that result with the third number.

Example: Find the HCF of 12, 18 & 24.

• HCF of 12 & 18 = 6

• HCF of 6 & 24 = 6

• So, HCF = 6
  
  

HCF of 4 Numbers

For four numbers, the process is the same.

• First, find the HCF of two numbers.

• Then, use that result with the next number.

• Finally, find the HCF of the fourth number.

Example: Find the HCF of 16, 24, 32, and 40.

• HCF of 16 & 24 = 8

• HCF of 8 & 32 = 8

• HCF of 8 & 40 = 8

• So, HCF = 8
  
  

HCF of Prime Numbers

Prime numbers are numbers that have only two factors: 1 and the number itself. Examples are 2, 3, 5, 7, 11, etc.

• If two different prime numbers are given, their HCF is always 1.

• If the same prime number is repeated, the HCF is that prime number.

Example:

• HCF of 5 and 11 = 1 (since they have no common factor except 1)

• HCF of 7 & 7 = 7
  
  

Properties

1. The HCF of two or more numbers is always a factor of each number.

2. The HCF is always less than or equal to the smallest number.

3. The HCF of two coprime numbers (numbers with no common factor except 1) is always 1.

4. If one number divides the other exactly, then the smaller number is the HCF.
   
   

Solved Examples

Example 1: Find the HCF of 18 and 24.

• Factors of 18 = 1, 2, 3, 6, 9, 18

• Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

• Common Factor = 1, 2, 3, 6

• The largest common factor is 6.

• So, HCF = 6

Example 2: Find the HCF of 20, 30, and 50.

• Factors of 20 = 1, 2, 4, 5, 10, 20

• Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30

• Factors of 50 = 1, 2, 5, 10, 25, 50

• Common Factors = 1, 2, 5, 10

• The largest common factor is 10.

• So, HCF = 10
  
  

Word problems

Problem 1: Two ropes are 12 meters and 18 meters long. What is the longest length of rope pieces that can be cut from both ropes without being left over?

Solution: We must find the HCF of 12 and 18.

• Factors of 12 = 1, 2, 3, 4, 6, 12

• Factors of 18 = 1, 2, 3, 6, 9, 18

• Common Factor = 1, 2, 3, 6

• HCF = 6

• So, the longest length of rope pieces is 6 m.

Problem 2: Three children have 24, 36, and 60 chocolates. They will divide the chocolate equally among themselves, without leaving anyone out. What is the maximum number of chocolates each child can get?

Solution:

We must find the HCF of 24, 36, and 60.

• Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

• Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

• Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

• Common Factor = 1, 2, 3, 4, 6, 12

• HCF = 12

• So each child can get 12 chocolates.
  
  

Practice Problems

1. Find the HCF of 28 & 42.

2. Find the HCF of 45, 60, and 75.

3. The lengths of two sticks are 32 cm & 48 cm. What is the longest piece of stick that can be cut from both without any leftover?

4. Three numbers: 18, 24 & 30. Find out their HCF.

5. A farmer has 20 apples, 30 mangoes, and 40 oranges. She wants to make fruit baskets with an equal number of each fruit. What is the maximum number of baskets she can make?
   
   

Frequently Asked Questions on HCF

1. What is the HCF of 36 and 24?

Answer: The HCF of 36 and 24 is 12.

• Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

• Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24

• Common divisors: 1, 2, 3, 4, 6, 12

• The highest common divisor is 12.

 

2. How to calculate HCF?

Answer: You can calculate the HCF by using any one of the following methods:

• Prime Factorization: Factorize every number into its prime factors and take the lowest powers of common factors multiplied.

• Division Method of HCF: Divide the bigger number by the smaller number and keep dividing the divisor by the remainder until the remainder becomes zero. The previous non-zero remainder is the HCF.

• Listing Divisors: Take all the divisors of both numbers and determine the greatest common divisor.

 

3. What is the HCF of 42 and 162?

Answer: The HCF of 42 and 162 is 6.

• Divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42

• Divisors of 162: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162

• Common divisors: 1, 2, 3, 6

• The highest common divisor is 6.

 

4. What is the HCF of 4 and 7?

Answer: The HCF of 4 and 7 is 1.

• Divisors of 4: 1, 2, 4

• Divisors of 7: 1, 7

• The only shared divisor is 1, so 4 and 7 are co-prime.