Least Common Multiple


Share |

The least common multiple (LCM) of a set of numbers is the smallest number that each of the original numbers is a factor of. Another way to say this is that the least common multiple of a numbers is the smallest number that can be divided evenly by all the numbers we are trying to find the LCM for.

To better understand what the least common multiple is, we will first look at sort of a brute force way of finding the LCM of two numbers. This method is effective and is probably the best way to do it for small numbers since we can do it easily in our heads, however when using larger numbers it may not always work so well.

Find the LCM of 5 and 8:

We can denote the LCM of these two numbers as LCM(5,8). In order to find LCM(5,8) using our brute force method, we can simply list out a bunch of multiples of 5 and 8:

5 x 1 = 5
5 x 2 = 10
etc.

8 x 1 = 8
8 x 2 = 16
etc.

So lets say we are going to start by listing out the first 10 multiples of 5 and 8:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

To find the Least Common Multiple of these two numbers we simply look for the first multiple that appears in both lists:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 8: 8, 16, 24, 32, 40,48, 56, 64, 72, 80

In this case our answer is 40. LCM(5,8) = 40

If you were to keep writing out the multiples of 5 eventually we would hit 5 x 16 = 80. While 80 is a common multiple of 5 and 8, it is not the Least Common Multiple.


 

Finding the Least Common Multiple Using Prime Factorization

Another more elegant way to find the least common multiple is to use prime factorization. Let's find the LCM(3,4,8).

The first step is to list out all of the prime factors of our numbers:

Prime Factors of 3: 3
Prime Factors of 4: 2 x 2
Prime Factors of 8: 2 x 2 x 2

We can now rewrite these factors using using exponent notation:

Prime Factors of 3: 20 x 31
Prime Factors of 4: 22 x 30
Prime Factors of 8: 23 x 30

It helps to write any factors that the other numbers have but the current number does not as that number to the 0 power. Since any number to the 0 power = 1, it does not change anything.

We now look at each factor/exponent and pick out the highest exponent for that factor among our lists:

Prime Factors of 3: 20 x 31
Prime Factors of 4: 22 x 30
Prime Factors of 8: 23 x 30

We can now take those values and multiply them together to get our Least Common multiple:

23 x 31 =  8 x 3 = 24


We now have our answer LCM(3,4,8) = 24.

We can check this against our brute force method:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 8: 8, 16, 24



 

A Short Cut for Finding the Least Common Multiple

There is a short cut to finding the least common multiple that is similar to finding the prime factors and multiplying them together. The trick is however, you can only find the LCM or two numbers at a time. 

Find LCM(6, 8)

To find the LCM of 6 and 8, we will first find the greatest common factor of 6 and 8:

GCF(6,8) = 2

We can then multiply the two numbers together and divide by the GCF:

(6 x 8)/2 = 48/2 = 24.

We now know that LCM(6,8) = 24.
 

Using the Short Cut to Find the LCM of 3 or More Numbers 

Let's say that we wanted to find LCM(3,6,8). Since we already know that the LCM(6,8) = 24, we know that the smallest LCM(3,6,8) can be is 24. To figure out LCM(3,6,8) we can just calculate LCM(3,24):

LCM(3,24) = (3 x 24)/(GCF(3,24)) = 72/3 = 24

Since LCM of (3, 24) = 24 and the LCM of (6,8) = 24 we know that the LCM(3,6,8)=24.



Math Mix iPhone App

Signup For Our Newsletter

Signup for our newsletter and get updates about all the great new features on MathMix.com. We will soon be adding more worksheets, puzzles, online practice and more!