## Least Common Multiples Of Numbers
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Hi there. This is a guide on least common multiples in mathematics. The student should be familiar with multiplication up to 12.


<center><img src="https://cdn.pixabay.com/photo/2019/04/04/03/54/pay-4101950_960_720.jpg" /></center>
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### Topics
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* What Is A Multiple?
* Count By Method To Find Least Common Multiples Of Two Numbers
* Extension: Least Common Multiples Of Three Numbers
* Answers To Practice Problems

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### What Is A Multiple?
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Given the number of items in one group, a multiple represents a total from a number of groups with each group being of a certain size. 

As an example some multiples of 6 are 6, 12, 18 and 24. The number six can be represented by 1 group of 6, 12 from 2 groups of 6, 18 from 3 groups of 6 and 24 being from 4 groups of 6.

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Visually, we have:

<center>1 x 6 = 6</center>

<center>2 x 6 = 12</center>

<center>3 x 6 = 18</center>

<center>4 x 6 = 24</center>

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or you can displaying this with a couny by method such as this:

<center>6, 12, 18, 24</center>

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### The Count By Method To Find Least Common Multiples Of Two Numbers
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It is best to show the count by method for finding the least common multiple with examples.


**Example One**

Find the least common multiple of 2 and 3.

Counting by 2 starting at 2 gives 2, 4, 6, 8, 10, 12, 14, 16, 18 and so on. With counting up by three starting at three consists of 3, 6, 9, 12, 15, 18. The least common multiple is the smallest number that both numbers share.

Common multiples of 2 and 3 are 6, 12 and 18. The least common multiple here is **6**.

**Example Two**

Find the least common multiple of 6 and 9.

Counting by 2 starting at 2 gives 2, 4, 6, 8, 10, 12, 14, 16, 18 and so on. With counting up by three starting at three consists of 3, 6, 9, 12, 15, 18. The least common multiple is the smallest number that both numbers share.

Common multiples of 2 and 3 are 6, 12 and 18. The least common multiple here is **6**.


**Practice Problems**

Determine the least common multiple for the pair of numbers below.

A1) 2 and 10

A2) 3 and 13

A3) 7 and 8

A4) 5 and 15

A5) 20 and 8

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<center><img src="https://cdn.pixabay.com/photo/2015/03/26/10/42/letters-691842_960_720.jpg" /></center>
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### Extension: Least Common Multiples Of Three Numbers
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A natural extension is going from finding least common multiples of two numbers to three numbers. It is still recommended to use the count by method for determining the multiples of the numbers.

**Example One**

Find the least common multiple of 2, 6 and 7.

Here are the multiples of 2, 6 and 7 (respectively).

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<center>2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, **42**</center>

<center>6, 12, 18, 24, 30, 36, **42**, 48, 54, 60</center>

<center>7, 14, 21, 28, 35, **42**, 49, 56</center>

The least common multiple of 2, 6 and 7 is 42 (in bold).

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**Practice Problems**

Determine the least common multiple for the triplet of numbers below.

B1) 2, 5 and 10

B2) 2, 3 and 6

B3) 3, 6 and 10

B4) 2, 7 and 8

B5) 2, 5 and 11

B6) 3, 10 and 15


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<center><img src="https://cdn.pixabay.com/photo/2019/02/22/19/03/digits-4014181_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2019/02/22/19/03/digits-4014181_960_720.jpg">Pixabay Image Source</a></center>


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### Answers To Practice Problems
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A1) 10

A2) 39

A3) 56

A4) 15

A5) 40

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B1) 10

B2) 6

B3) 30

B4) 56

B5) 110

B6) 30

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