In mathematics, a multiple is a product of any number and an integer. The numbers 16, -48 and 72 are multiples of 8 because 8 x 2 = 16, 8 x -3 = -48 and 8 x 9 = 72. Similarly, the first five positive  multiples of 7 are the following:

7, 14, 21, 28, 35.

In this post, we will particularly talk about positive integers and positive multiples.  This is in preparation for the discussions on addition and subtraction of fractions.

We can always find a common multiple given two or more numbers. For example, if we list all the positive multiples of 2 and 3, we have

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

and

3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

As we can see, in the list, 6, 12 and 18 are common multiples of 2 and 3. If we continue further, there are still other multiples, and in fact, we will never run out of multiples.

Can you predict the next five multiples of 2 and 3 without listing?

The most important among the multiples is the least common multiple.  The least common multiple is the smallest among all the multiples. Clearly, the least common multiple of 2 and 3 is 6. Here are some examples.

Example 1: Find the least common multiple of 3 and 5

Multiples of 3: 3, 6, 9. 12, 15, 18

Multiples of 5: 5, 10, 15, 20, 25,30

As we can see, 15 appeared as the first common multiple, so 15 is the least common multiple of 3 and 5.

Example 2: Find the least common multiple of 3, 4, and 6.

In this example, we find the least multiple that are common to the three numbers.

Multiples of 3: 3, 6, 9, 12, 15

Multiples of 4: 4, 8, 12, 16, 20

Multiples of 6: 6, 12, 18, 24, 30

So, the least common multiple of 3, 4, and 6 is 12.

Example 3: Find the least common multiple of 3, 8 and 12.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24

Multiples of 8: 8, 16, 24,

Mulitples of 12: 12, 24, 36, 48, 60

So, the least common multiple of 3, 4 and 6 is 24.

In the next part of this series, we will discuss about How to Add Fractions.