What is LCM?

What is LCM?

Least Common Multiple (L.C.M.) 

 

The multiples which are common to a given set of numbers are called their common multiples. 

For example, the common multiples of 6 and 8 are 24, 48, 72, 96 and so on. 

The lowest among these common multiples is 24 which is the Lowest Common Multiple of 6 and 8. 

Thus, the multiple which is smallest among the common multiples of a given set of numbers is called their Lowest Common Multiple (L.C.M.). 

 

Example- Let us take 2 numbers 12 and 18 and find their LCM. 

Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, ... 

Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, ... 

Common multiples of 12 and 18 are 36, 72, 108 and so on. 

Smallest among all common multiples is 36. 

Thus, L.C.M. of 12 and 18 = 36 

 

 

  • LCM byPrime Factorization Method 

 

  • In this method, we express the given set of numbers as a product of their prime factors. 

 

  • The LCM of the twonumbers is calculated as the product of the prime factors counted the maximum numberof times they occur in any of the numbers. 

 

Example-  

  1. Find the LCM of 18 and 24. 

 

Let us find the prime factorization of 18 and 24. 

 

 

So, 18 = 2 × 3 × 3 

      24 = 2 × 2 × 2 × 3 

Maximum number of times prime factor 2 occurs among 18 and 24 is 3. 

Similarly, 3 occurs maximum of 2 times. 

LCM of 18 and 24 = 2 × 2 × 2 × 3 × 3  = 72 

Thus, LCM (18 and 24) = 72 

 

Relationship between HCF and LCM 

 

The product of HCF and LCM of two numbers is equal to the product of the numbers. 

 

i.e. HCF × LCM = Product of the Numbers 

 

Example- Given two numbers: 6 and 8 

HCF (6 and 8) = 2 

LCM (6 and 8) = 24 

Product (6 and 8) = 6 × 8 = 48 

Now, HCF × LCM = 2 × 24 = 48  

 

Thus, HCF × LCM = Product of the Numbers. 

Can you solve this?? 

#Find HCF and LCM of the form- 2222....30 times and 3333....70 times (any random numbers) 

 

Solution: 

 

HCF  

  • Take HCF of no. that is being repeated...i.e. HCF of 2 & 3.  
    i.e. 1  

  • Take HCF of no. of time these nos. are being repeated i.e. HCF of 30 and 70 is 10.  

  • So, the HCF of the two given numbers is 1...written 10 times. 

 

LCM  

  • Take LCM of no. that is being repeated...i.e. LCM of 2 & 3.  
    i.e. 6  

  • Take LCM of no. of time these nos. are being repeated i.e. LCM of 30 and 70 is 210.  

  • So, the LCM is 6...written 210 times 

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