An Understanding About LCM and HCF

To learn about the concepts of the Highest Common Factor [HCF] and Lowest Common Multiple [LCM], we would first understand what is a factor or a multiple.  

Each number that can divide a particular number completely without leaving any remainder is known as a factor of the number. For example, we will take the number 12. 

12 is completely divisible by the numbers 1, 2, 3, 4, 6, 12. Hence these are the factors of the number 12. Now we will take a look at LCM and HCF and understand both.  

What is LCM? 

The smallest or the least number which is exactly divisible by each number in the set of the numbers present is called the lowest common multiple or LCM. For example, let’s take the numbers 6 and 8. 

Now, in order to find the LCM of two numbers we will look at their multiples: 

Multiples of 6 are: 

6 X 1 = 6  

6 X 2 = 12 

6 X 3 = 18 

6 X 4 = 24, and so on…. 

Multiples of 8 are:  

8 X 1 = 8 

8 X 2 = 16 

8 X 3 = 24 

8 X 4 = 32, and so on… 

Here, we can see that 24 is the lowest common multiple of the two numbers and hence 24 is the LCM. 

What is HCF? 

The biggest or the largest number which can be each number in a set of numbers is known as the highest common factor or the HCF. For example, let’s take the numbers 14 and 16. 

Factors of 14 are: 

1 X 14 = 14 

2 X 7 = 14 

7 X 2 = 14 

Factors of 16 are:  

1 X 16 = 16 

2 X 8 = 16 

4 X 4 = 16 

8 X 2 = 16 

Here, we can see that the common factors of 14 and 16 are 1 and 2 respectively. As 2 is higher in numerical value 2 will be considered as the HCF of the two numbers.  

LCM and HCF formula 

The formula of LCM and HCF is a mathematical computation of the relation between LCM and HCF. It is fairly simple to understand, the LCM and HCF formula is as follows: 

Product of two numbers = HCF of two numbers X LCM of the two numbers.

Let’s say the two numbers are y and z so according to the LCM and HCF formula: 

y X z = HCF [y, z] X LCM [y, z]  

Looking at the relationship between LCM and HCF we can derive the following formulae: 

LCM of two numbers: product of the numbers / HCF of the two numbers {/ = divided} 

HCF of two numbers: product of the numbers / LCM of the two numbers {/ = divided} 

How to find LCM and HCF? 

There are two ways to find the LCM and HCF of two numbers respectively. They are: 

• Prime factorization method  
• Division method 

We will look at the prime factorization method:  

HCF by Prime factorization method:  

To find the HCF by the prime factorization method, we will first find the prime factors of the two numbers. Then we will look at the common prime factors of each of the numbers. Multiplying the common prime factors of the numbers will give us the HCF. 

Let’s take the example of 25 and 50 

Prime factors of 25 are: 5 X 5  

Prime factors of 75 are: 2 X 5 X 5 

The common prime factors are 5 X 5 and hence 25 becomes the HCF of [25, 75] 

LCM by prime factorization method:  

To find the LCM of two numbers by the prime factorization method, we will find the prime factors of each of the numbers. After finding the common prime factors of each number, we will multiply them by the uncommon prime factors of each number, this process will give us the LCM.  

Let’s take the example of 75 and 90 

Prime factors of 75 are: 2 X 5 X 5 

Prime factors of 90 are: 2 X 3 X 3 X 5 

Now, LCM = [ 2 X 5] X [ 5 X 3 X 3] = 

 10 X 45 = 450 is the LCM of the two numbers.  

Conclusion 

Thus with the help of this article, we can safely conclude that HCF stands for the highest common factor whereas LCM stands for the lowest common multiple. HCF is always the biggest or the largest of all common factors present whereas LCM is always the smallest or the lowest of all the common multiples present.  

HCF of any set of numbers can never be greater than the numbers themselves whereas LCM of any set of numbers can never be smaller than the numbers themselves. Be sure to conduct your own research to learn more about the topic of LCM and HCF.