LOWEST COMMON MULTIPLE

 L.C.M. stands for Lowest Common.

LCM

So let us first discuss about what are multiples of the given number.

Multiple: The number obtained by multiplying the given number is called the multiple of given number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Lowest Common Multiple: The L.C.M. of two or more given numbers is the lowest (smallest) number that is a multiple of each of the given numbers.

Thus, it is the smallest number which is exactly divisible by each of the given numbers.

For example: L.C.M. of 4 and 6 is 12 as 12 is the least multiple common multiple of 4 as well as 6.

Methods of finding L.C.M.: The following three methods are most commonly used to find the L.C.M.

1) Common Multiple Method:

Step 1: Find a few multiples of each given number.

Step 2: From the multiples obtained in Step 1, select the common ones.

Step 3: The lowest number (multiple) obtained in Step 2 is the required lowest common multiple of the given numbers.

For example: Find L.C.M. of 4, 5 and 10.

Step 1: Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ......

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45 ,50, 55, .....

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, .......

Step 2: Common multiples of 4, 5 and 10 = 20, 40, 60, 80, .....

Step 3: smallest common multiple obtained in step 2 is 20.

So, L.C.M. of 4, 5 and 10 is 20.

2) Prime Factor Method: 

Step 1: Express each of the given number as a product of its prime factors and then in index form.

Step 2: Product of all the prime factors obtained in step 1 with highest power of each is the L.C.M. of given numbers.

For example: Find L.C.M. of 18 , 24 and 36.

Step 1: 18 = 2 × 3 × 3 = 2¹ × 3²,

              24 = 2 × 2 × 2 × 3 = 2³ × 3¹,

              36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 2: Since, the prime factors 2 and 3, obtained above, with highest power are 2³ and 3².

Therefore,  L.C.M. = 2³ × 3² = 2 × 2 × 2 × 3 × 3 = 72.

3) Common Division Method: 

Step 1: Write all the given numbers in a horizontal line, separating them by commas.

Step 2: Divide by a suitable number, that exactly divides at least two of the given numbers. And, write down the quotients and the undivided numbers obtained, below the first line.

Step 3: Repeat the process until we get a line of numbers that are prime to one-another.

Step 4: The product of all the divisors and the numbers obtained in the last line will be the required L.C.M.

For example: Find L.C.M. of 16, 20 and 24.

        16, 20, 24

    2  16, 20, 24

        8, 10, 12

     2 16, 20, 24

     2  8, 10, 12

     2  4, 5, 6

         2, 5, 3

     Therefore, L.C.M = 2 × 2 × 2 × 2 × 3 × 5 = 240.