# Prime and Composite Numbers

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We are now familiar with the **factors** of a number.

Observe the number of factors of a few numbers arranged in this table.

Numbers | Factors | Number of Factors |
---|---|---|

1 | 1 | 1 |

2 | 1, 2 | 2 |

3 | 1, 3 | 2 |

4 | 1, 2, 4 | 3 |

5 | 1, 5 | 2 |

6 | 1, 2, 3, 6 | 4 |

7 | 1, 7 | 2 |

8 | 1, 2, 4, 8 | 4 |

9 | 1, 3, 9 | 3 |

10 | 1, 2, 5, 10 | 4 |

11 | 1, 11 | 2 |

12 | 1, 2, 3, 4, 6, 12 | 6 |

We find that (a) **The number 1** has only one *exactly two factors* **1** and the number itself. Such number are **2, 3, 5, 7, 11** etc. These numbers are

Try to find some more **prime numbers** other than these.

(c) There are numbers having more than two *factors* like **4, 6, 8, 9, 10** and so on.

These numbers are

Is 15 a composite number? Why? What about 18? 25?

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Without actually checking the factors of a number, we can find prime numbers from 1 to 100 with an easier method.

## The Sieve of Eratosthenes

It turned out to be quite difficult to determine if a number is prime: you always had to find *all* its prime factors, which gets more and more challenging as the numbers get bigger. Instead, the Greek mathematician **Sieve of Eratosthenes**.

**2**. Any multiple of 2 can’t be prime, since it has 2 as a factor. Therefore we can cross out all multiples of 2.

**3**– again a prime number. All multiples of 3 can’t be prime, since they have 3 as a factor. Therefore we can cross these out as well.

**5**: it is a prime number and again we cross out all multiples of 5.

Now we can count that, in total, there are

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1.**Write all the prime numbers less than 15.**

By observing the Sieve Method, we can easily write the required prime numbers as

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**Observe that 2 × 3 + 1 = 7 is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?**

2 × 5 + 1 =

2 × 8 + 1 =

2 × 9 + 1 =

2 × 15 + 1 =