The Least Common Multiple of two or more rational numbers is the smallest positive rational number which is a multiple of all the given rational numbers.

For example, let us find the LCM of 2 and 6.
Multiples of 2 are 2, 4, 6, 8, 10, 12.....
Multiples of 6 are 6, 12, 18, 24, 30....

From the numbers listed above, we can easily find out that 6 is the least common multiple of 2 and 6.

LCM can be defined as the smallest of the common multiples. Also, a common multiple of two numbers that is exactly divisible by each number. Hence, we can define the least common multiple(LCM) of two numbers as the smallest number, which is not equal to zero, which is a multiple of both.

Let us consider the multiples of some numbers. The multiple of a number can be taken as those numbers obtained by multiplying it by other numbers.

Multiples of 4 are 4, 8, 12, and so on.
Multiples of 6 are 6, 12, 18 and so on.

From these two groups, it can be seen that, 12 is common to both or it can be said that 12 is a common multiple of 6 and 4. There are several common multiples of 6 and 4. They are 12, 24, 36 and so on. The least of the common multiples for the above two numbers is 12. Hence, 12 is the least common multiple of 6 and 4.

There are a few steps to be followed to find the Least Common Multiple of Numbers:

  • Find the Greatest Common Divisor (GCD) of the numbers
  • Multiply the numbers together
  • Divide the product of the numbers by the GCD.
LCM of two prime numbers is the product of those two prime numbers.

Examples on Finding Least Common Multiple:

Given below are some examples that explain how to find the lcm of given numbers:

Example 1:

Find the least common multiple of 3 and 4.

Solution:

Multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24, ................
Multiples of 4 are : 4, 8, 12, 16, 20, 24, 28, .....................
Therefore, the least common multiple of 3 and 4 = 12.

Example 2:

Find the least common multiple of 2, 3 and 6

Solution:

Multiples of 2 are : 2, 4, 6, 8, 10, 12, 14, 16, 18,................
Multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24, ...............
Multiples of 6 are : 6, 12, 18, 24, 30, 36, ..................
So, the least common multiple of 2, 3 and 6 = 6

The common denominator may be any number of which each given denominator is a factor or any number which could be divided by each of them without a remainder. Such numbers are termed as common multiples of all common divisors. The least number that will accomplish this is called their least common multiple.

The least common denominator would be considered for any fraction as the least multiple of all their denominators.

Here is an example on finding the least common denominator.

Find the least common denominator of 4 and 6. 24 is obviously a common multiple of 4 and 6, for this measure exactly for both of them, but 12 will do the same for 24 and so it will be the least number that will do this. So, 12 becomes the least common multiple for both 4 and 6.

The factors for 4 are: 2 x 2 and the factors for 6 are: 3 x 2.
The least common multiple of 4 and 6 would be: 2 x 2 x 3 = 12
Therefore, 12 becomes the least common denominator.

Given below are some examples on Least Common Multiple(LCM).

Example 1:

Find the least common multiple of 4 and 6

Solution:

Multiples of 4 are : 4, 8, 12, 16, 20, 24, 28, .....................
Multiples of 6 are : 6, 12, 18, 24, 30, 36, ............................
So, the least common multiple 12 is exactly divisible by both 4 and 6.
Therefore, the least common multiple of 4 and 6 = 12.

Example 2:

Find the least common multiple of 2, 3 and 5

Solution:

Multiples of 2 are : 2, 4, 6, 8, 10, 12, 14, 16, 18,20,22,24,26, 28,30,32.......
Multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24, 27,30,33,36......
Multiples of 5 are :5,10,15,20,25,30,35,40,... .......,
So, the least common multiple 30 is exactly divisible by 2, 3 and 5.
Therefore, the least common multiple of 2, 3 and 5 = 30.

Given below are practice problems on least common multiple.

Practice Problem 1:

Find the LCM of 16, 18 and 20

Practice Problem 2:

Find the LCM of 9, 12 and 27