Relation Between HCF and LCM – Applications and Examples

What is the relation between HCF and LCM?

Before coming to the relation between these two, let us have a quick recap of LCM and HCF.

What is HCF (Highest Common Factor)?

HCF is the highest common factor of the given numbers or the largest number, which divides the given numbers completely without leaving any remainder.

For example, to find the HCF of 6 and 10, we must find the largest number that can divide 6 and 10 wholly.

6 is wholly divisible by 1, 2, 3, and 6.

10 is wholly divisible by 1, 2, 5, and 10.

Here, 2 is the largest common number that can divide 6 and 10 altogether. 

So, the required HCF = 2.

What is LCM (Least Common Multiple)?

As the full form of LCM suggests, it is the least (smallest) common multiple of the given numbers.

For example, to find the LCM of 6 and 10, we will have to find the smallest common multiple of the given numbers.

The first five multiples of the number 6 are 6, 12, 18, 24, and 30.

The first five multiples of the number 10 are 10, 20, 30, 40, and 50.

30 is the smallest number that is common in both. 

So, the LCM of the given numbers is 30.

Relation of HCF and LCM

(i) The product of HCF and LCM of two numbers is always equal to the product of the given numbers.

Means, (HCF × LCM) of two numbers = Product of the numbers.

Example:  Let us verify the above relation by taking two numbers, 10 and 15.

Solution:

HCF of 10 and 15 = 5

LCM of 10 and 15 = 30

LCM × HCF = 30 × 5 = 150

Product of the given numbers = 10 × 15 = 150

Hence, it is verified that HCF × LCM of two numbers = Product of the numbers.

 Note- Remember that this rule is applicable only for two numbers. The product of HCF and LCM of three numbers is never equal to the product of the given numbers.

(ii) For co-prime numbers, HCF is 1, and LCM is the product of the numbers.

Co-prime numbers are the pair of numbers whose common multiple is only and only 1. 

Example: Let us verify the above relation by taking co-prime numbers, 7 and 11. 

Solution:

HCF (7 and 11) = 1

LCM (7 and 11) = 77

Product of the given numbers = 7 × 11 = 77

Thus, we verified that the HCF of co-prime numbers is 1 and LCM = Product of the Numbers.

Examples 

1. The HCF of the two numbers is 12, and their LCM is 80. If one of the two numbers is 24, find the second number.

As per the property of HCF and LCM of two numbers- 

LCM × HCF = Product of the numbers

Here, LCM = 80, HCF = 12 

So, 80 × 12 = 24 × Other Number

Therefore, Other Number = (80 × 12) / 24 = 80/2 = 40

2. If the product of two co-prime numbers is 2021, find the LCM of the numbers.

According to the property, LCM of Co-prime Numbers = Product Of the Numbers.

So, LCM of Co-prime Numbers = 2021.