LCM Full Form - How to find LCM, Common listing, Prime factorization and Division Methods

LCM stands for least common multiple. It is the smallest common factor between two or more numbers. This means the smallest number divisible by all the original numbers is said to be the least common multiple of those numbers.

For example, we need to find the least common multiple of the numbers 5 and 6.
The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, and so on.
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, and so on.
The smallest common multiple we can see common in both the numbers is 30. Therefore, 30 is the LCM of 5 and 6.

How to find LCM?

1. Common listing method: - In this method, we need to list down all the possible multiples and find the smallest common between them. This method is lengthy and time taking. It is suitable for small numbers.
   
   For example, let us find the LCM of numbers 3 and 7.
   Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 and so on.
   Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 and so on.
   We can see the smallest common number between 3 and 7 is 21. Therefore, 21 is the LCM of 3 and 7. We can also see that this method is time taking if done for more than two numbers.


2. Prime factorization method: - This method is suitable for both large numbers and small numbers both. In this method, we need to find the common multiple between numbers (or the smallest prime multiple), count them once, and multiply the remaining numbers.
   
   For example, let us find the LCM of numbers 15 and 21 by the prime factorization method.
   Prime factors of 15 are 3 x 5
   Prime factors of 21 are 3 x 7
   In these two prime factors, we find 3 as a common multiple and 5, 7 remain as it is. Therefore, we will count 3 only once and will take 5 and 7 separately.
   This implies, the LCM of 15 and 21 is 3 x 5 x 7 = 105.
   We can check this by the common listing method as well.


3. Division method: - This method is the best suitable for all numbers, two or more than two. All we need to do is find the common prime factor between those numbers and start dividing. The remainder should be written, and further division should continue.
   
   For example, let us find the LCM of 15, 21, and 30 by the division method.
   The smallest common prime factor between these three is 2. 15 and 21 are not divisible by 2, so we will keep them as they are. We will divide these by 2, and we get 15, 21, and 15 as remainders. Remember, we need to divide them only when they are completely divisible by the smallest prime factor.
   
   Next, the smallest common prime factor between 15, 21, 30 is 3. Upon dividing, we will get 5, 7, 5 as the remainder. Repeating this process we will get LCM as 2 x 3 x 5 x 7 = 210.