**What is a Multiple?**

A multiple is a result that is obtained by multiplying a number by an integer( it shouldn’t be a function ). The multiples of the whole number are obtained by taking out the product of any of the counting numbers and that of whole numbers.For example, To find the multiples of 6 we multiply 6 by 1, 6 by 2, 6 by 3, and so on. The multiples are the product of this multiplication.

Example 1: | Find the multiples of whole number 3 | |||||||

Multiplication: | 3 x 1 | 3 x 2 | 3 x 3 | 3 x 4 | 3 x 5 | 3 x 6 | 3 x 7 | 3 x 8 |

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

Solution: | The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24 … |

Example 1: | Find the multiples of whole number 8 | |||||||

Multiplication: | 8 x 1 | 8 x 2 | 8 x 3 | 8 x 4 | 8 x 5 | 8 x 6 | 8 x 7 | 8 x 8 |

Multiples of 8: | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 |

Solution: | The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64 … |

**Common Multiples**

Multiples that are common to two numbers are known as common multiples of those numbers. Let us understand with the help of an example.

Consider two numbers– 30 and 45. Multiples of 30 and 45 are –

**30 = 30, 60, 90, 120, 150, 180, 210, 240, 270…..**

**45 = 45, 90, 135, 190, 225, 270……….**

We see that **90** and **270** are first two common multiples of 30 and 45. But what can be the real life use of common multiples?

Suppose Joe and Sam are running on a circular track. They start from the same point but Joe takes 30 second to cover a lap while Sam takes 45 second to cover the lap. So when will be the first time they meet again at the starting point?

This can be deduced from the list of common multiples. Sam and Joe will meet again after 90 minutes.

**Common Factors**

Factors that are common to two or more numbers are known as their common factors. So how to find the common factors? Consider two numbers A and B. Write down all the factors of A and B separately and observe the numbers which are common to both A and B. Let us understand the concept of common factors with the help of an example.

Considering the same two numbers 30 and 45.

Factors of 30 and 45 are –

**30 = {1, 2, 3, 4, 5, 6, 12, 15, 30} **

**45 = {1, 3, 5, 9, 15, 45}**

What are the common factors that you can observe? 15, 5, 3 and 1 appears in both 30 and 45.

**Common factors of 30 and 45 are 1, 3, 5 and 15.**

This concept has lots of practical application too. Suppose you want to floor a room of dimension 30 m x 45 m. The maximum size of a square tile that can be used can be deduced with the help of common factors. As 15 is the highest common factor, a square tile of side length 15 cm should be used.

**Co-prime Number**

Two numbers having only ‘1’ as their common factor are known as co-prime numbers. Example: 5 and 14, 6 and 17.

**Note:**

- Two prime numbers are always co-prime numbers.
- Two even numbers can never be co-prime numbers.

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