# Least Common Multiple DefinitionSupport Page

Welcome to our Least Common Multiple Definition page.

As well as a clear definition of the least common multiple we have links to our Least Common Multiple calculator and also our Least Common Multiple worksheets pages.

## Least Common Multiple Definition

### Finding the Least Common Multiple (or LCM) - by Listing the Multiples Method

#### Examples

Example 1) Find the least common multiple of 5 and 3.

The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ...

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

So we have found two common multiples which occur on both lists: 15 and 30.

Note that we really only needed to go up to the 3rd multiple of 5, or the 5th multiple of 3 to find our answer.

The Least Common Multiple is 15 (which is 3 x 5).

Example 2) Find the lowest common multiple of 5 and 11.

The multiples of 5 are: 5, 10, 15, 20 , 25, 30, 35, 40, 45, 50, 55, 60

The multiples of 11 are: 11, 22, 33, 44, 55, 66, ...

The smallest number on both lists is 55 (which is 5 x 11).

The Lowest Common Multiple is 55.

Example 3) Find the lowest common multiple of 6 and 15.

The multiples of 6 are: 6, 12, 18, 24 , 30, 36, 42, 48, 54, 60, 55, 60

The multiples of 15 are: 15, 30, 45, 60, 75, 90, ...

We have found two common multiples: 30 and 60.

The smallest number on both lists is 30 (note that this is a smaller value than 6 x 15).

The Lowest Common Multiple is 30.

Example 4) Find the least common multiple of 4 and 6 and 10.

The multiples of 4 are: 4, 8, 12, 16 , 20, 24, 28, 32, 36, 40, 42, 48, 52, 56, 60, 64

The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66...

The multiples of 10 are: 10, 20, 30, 40, 50, 60, ...

The smallest number on all the lists is 60.

The Least Common Multiple is 60.

### Least Common Multiple Calculator - The Prime Factors Method

Example 1) Find the lowest common multiple of 14 and 20.

As a product of prime factors:

• 14 = 2 x 7
• 20 = 2 x 2 x 5 = 22 x 5

Next combine the factors and sort this list by order of number and exponent:

22, 2, 5, 7

Now remove any lower exponents of the same number from the list.

The factor 2 is repeated so we only need to use the highest exponent which is 22.

Our new list is:

22, 5, 7.

Now we multiple the factors of the list together.

This gives us: 22 x 5 x 7 = 2 x 2 x 5 x 7 = 140

The Least Common Multiple is 140.

Example 2) Find the least common multiple of 144 and 78.

As a product of prime factors:

• 144 = 2 x 2 x 2 x 2 x 3 x 3 = 24 x 32
• 78 = 2 x 3 x 13

Next combine the factors and sort this list by order of number and exponent:

24 x 2 x 32 x 3 x 13

Now remove any lower exponents of the same number from the list.

The factor 2 is repeated so we only need to use the highest exponent which is 24.

The factor 3 is repeated so we only need to use the highest exponent which is 32.

This gives us a final list of:

24, 32, 13.

Now we need to multiple the prime factors together to find the LCM.

This gives us: 24 x 32 x 13 = 2 x 2 x 2 x 2 x 3 x 3 x 13 = 1872

As you can probably see, this would take a while to work out using the multiples method!

The Least Common Multiple (LCM) is 1872.

Example 3) Find the least common multiple of 14, 31 and 12.

As a product of prime factors:

• 14 = 2 x 7
• 31 = 31 (it is already prime)
• 12 = 2 x 2 x 3 = 22 x 3

Next combine the factors and sort this list by order of number and exponent:

22, 2, 3, 7, 31

The factor 2 is repeated and we only need to use the highest exponent which is 22.

This gives us a final list of:

22, 3, 7, 31

Now we need to multiple the prime factors together to find the LCM.

This gives us: 22 x 3 x 7 x 31 = 2 x 2 x 3 x 7 x 31 = 2604

The Least Common Multiple (LMC) is 2604.

Example 4) Find the smallest common multiple of 129 and 321.

As a product of prime factors:

• 129 = 3 x 43
• 321= 3 x 107

Next combine the factors and sort this list by order of number and exponent:

3, 3, 43, 107

The factor 3 occurs in both numbers, and the exponent is the same, so we just need to include it once.

This gives us a final list of:

3, 43, 107

Now we need to multiple the prime factors together to find the LCM.

This gives us: 3 x 43 x 107 = 13803

The Least Common Multiple (LCM) is 13803.

### More Recommended Math Worksheets

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