What is Least Common Multiple [LCM]

 We can find the least common multiple of some numbers in several ways.First way by multiplication  method.        

Let us find the [LCM] of 2,3 and 4 by multiplication method

multiples of 2   -  2,4,6,8,10,12,14,16,18,20,22,24,26

multiples of 3  -  3,6,9,12,15,18,21,24

multiples of 4  -  4,8,12,16,20,24,28

12,24 are the common multiples of 2,3 and 4

Further, we can continue the common multiples of 2,3 and 4

12 is the first common multiple of 2,3 and 4

Then the [LCM] of 2,3 and 4 is 12

 

Second way is considering their prime factors

Let us find the [LCM] of 4,12 and 8 by considering their prime factors

 

4 = 2 * 2         =  2²

12 = 2 * 2 * 3    = 2² * 3¹

18 = 2 * 3 * 3   = 2¹ * 3²

  Let us select the greatest power of each prime factor

 

 There are two distinct prime factors, namely, 2 and 3. When the factors of all three numbers are considered.

 

 The power of 2 with the largest index = 2²

vThe power of 3 with the largest index = 3²

The [LCM] of the given numbers is the product of these greatest powers

Therefore, the [LCM] of 4,12 and 18

= 2² * 3²

^{=2 * 2 * 3 * 3}

⁼³⁶

^{Then the [LCM] of 4,12 and 18 is 36}

 

^{Third way is find the [LCM] by the division method}

^{Let us find the [LCM] of 4,12 and 18 by the division method}

 

^{write these numbers as shown}    

^{2  -  4   12   18}

^{2 -  2      6      9}

^{3 -  1      3      9}  

^{1      1     3}

^{Since all these numbers are divisible by 2, divide each of them by 2 individually.} 

 

^{We get 2, 6 and 9 as the result. No prime number divides all of them without remainder. However, 2 divides both 2 and 6 without remainder. Divide 2 and 6 by 2, and write the results below the respective numbers. Write 9 below 9.}

 

^{Since 3 and 9 are divisible by 3, the next smallest prime number, divide them by 3 and write the results below the respective numbers. Now observe that we cannot find at least two numbers which are divisible by the same number. Therefore, the division is stopped here.}

 

^{Multiply all divisors and all numbers left in the last row. The product gives the LCM of the given numbers.} 

^{Accordingly, the LCM of 4,12 and 18 = 2 * 2 * 3 * 1 * 1 * 3  =  36}

 

^{Then the [LCM] of the 4,12 and 18 is 36}