HCF and LCM - Definition, Formula, Full Form, Examples

Lesson Overview

• HCF and LCM Definition
• Difference Between HCF and LCM
• HCF and LCM Formula
• Methods To Find HCF and LCM
• HCF and LCM Examples

The HCF and LCM are tools of number theory. They help us understand how numbers are connected and have applications throughout mathematics.

The concepts of HCF and LCM are essential for simplifying fractions, solving problems, and working with algebraic expressions.

HCF and LCM Definition

HCF definition

The Highest Common Factor (HCF) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.

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. 
Here's how:
Prime factorization of 12:
12 = 2² ×3 

1. Prime factorization of 18:
   18 = 2 × 3²
2. Identify the common factors:
   The common prime factors are 2 and 3.
3. Take the lowest power of the common factors:
   • The lowest power of 2 is 2¹.
   • The lowest power of 3 is 3¹.
4. Multiply the common factors:
   2¹ x 3¹ = 6

So, the HCF of 12 and 18 is 6.

LCM definition

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of each of the integers.  

Example:

The LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 and 6.

Here's how

1. Prime factorization of 4:
   4=2²
2. Prime factorization of 6:
   6 = 2 x 3
3. Identify the unique prime factors:
   The unique prime factors are 2 and 3.
4. Take the highest power of each prime factor:
   • The highest power of 2 is 2².
   • The highest power of 333 is 3¹.
5. Multiply the highest powers:
   2² x 3¹ = 4 x 3 = 12

So, the LCM of 4 and 6 is 12.

Difference Between HCF and LCM

HCF and LCM, though related, have distinct properties and applications.

HCF and LCM Formula

The HCF and LCM are related by a formula that allows us to calculate one if we know the other and the product of the two numbers.

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

Methods To Find HCF and LCM

To find HCF and LCM, we use two methods- Prime Factorization Method and the Division Method. 

1. Prime Factorization Method
   

How to find HCF:

• Find the prime factorization of each number.
• Identify the common prime factors.
• Multiply the common prime factors, using the lowest power of each.
  

Example: Find HCF(36, 48)

• 36 = 2² x 3²   
• 48 = 2⁴ x 3¹
• HCF(36, 48) = 2² x 3¹ = 12  
  

How to find LCM:

• Find the prime factorization of each number.
• Identify all prime factors present in either number.
• Multiply all prime factors, using the highest power of each.

Example: Find LCM(12, 18)

• 12 = 2² x 3¹
• 18 = 2¹ x 3²
• LCM(12, 18) = 2² x 3² = 36

Take this quiz -

HCF And LCM Quiz

2.  Division Method

How to find HCF:

• Divide the larger number by the smaller number.
• If there is a remainder, divide the smaller number by the remainder.
• Continue this process until the remainder is 0.
• The last divisor is the HCF.   

Example: Find HCF(72, 126)

HCF = 18

How to find LCM:

• Write the numbers in a row.
• Divide by a prime number that divides at least two of the numbers.
• Write the quotients and any undivided numbers in the next row.
• Repeat steps 2 and 3 until no two numbers have a common prime factor.
• The LCM is the product of all the divisors and the numbers in the last row.
  

Example: Find LCM(12, 15, 20)

LCM(12, 15, 20) = 2 x 2 x 3 x 5 = 60

Take this quiz -

Shortcut of LCM and HCF Quiz

HCF and LCM Examples

1. Find the HCF of 72 and 90, through Prime Factorization.

• 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²
• 90 = 2 x 3 x 3 x 5 = 2¹ x 3² x 5¹
  

Both numbers have 2 and 3² as common prime factors.

HCF(72, 90) = 2¹ x 3² = 18

2. Find the LCM of 24 and 36, through Prime Factorization.

• 24 = 2 x 2 x 2 x 3 = 2³ x 3¹
• 36 = 2 x 2 x 3 x 3 = 2² x 3²
  

The prime factors involved are 2 and 3.

LCM(24, 36) = 2³ x 3² = 72

3.  Find the HCF of 56 and 84.

The last divisor is the HCF, so HCF(56, 84) = 28

4. Find the LCM of 15, 20, and 25.

LCM(15, 20, 25) = 5 x 3 x 5 x 2 x 2 = 300

5. The LCM of two numbers is 120, and their product is 480. Find their HCF.

• HCF x LCM = Product of two numbers
• HCF x 120 = 480
• HCF = 480 / 120 = 4

Take this quiz -

LCM & HCF Practice Set for MBA

6. The HCF of two numbers is 12, and their product is 360. Find their LCM.

• HCF x LCM = Product of two numbers
• 12 x LCM = 360
• LCM = 360 / 12 = 30

Take this quiz -

HCF And LCM Quiz Questions For Class 5