# Prime numbers

A natural number that is divisible only by one and itself is called a prime number.

## Introduction

The prime numbers are natural numbers but all natural numbers are not prime numbers. There are two different theories to determine the prime numbers from the natural numbers.

1. Division method
2. Factoring method

Now, let’s study both methods to learn how to determine the prime numbers from natural numbers.

#### Division method

There are some natural numbers, which are only divisible by one and themselves. They are called as prime numbers but the remaining natural numbers are also divided by one or more other natural numbers. So, they cannot be called as prime numbers.

### Example

$5$ and $6$ are two natural numbers.

$(1) \,\,\,\,\,\,$ $\dfrac{5}{1} \,=\, 5$

$(2) \,\,\,\,\,\,$ $\dfrac{5}{5} \,=\, 1$

The natural number $5$ is a prime number because it is divided by $1$ and $5$ but cannot be divided by any other natural number.

$(1) \,\,\,\,\,\,$ $\dfrac{6}{1} \,=\, 6$

$(2) \,\,\,\,\,\,$ $\dfrac{6}{6} \,=\, 1$

It seems, the natural number $6$ is a prime number but it is not because it can be divided by some other natural numbers $2$ and $3$ as well.

$(3) \,\,\,\,\,\,$ $\dfrac{6}{2} \,=\, 3$

$(4) \,\,\,\,\,\,$ $\dfrac{6}{3} \,=\, 2$

Therefore, $5$ is a prime number but $6$ is not a prime number.

#### Factorization method

Every natural number can be written as two or more factors. If a natural number has only two factors, then it is called as a prime number.

### Example

$7$ and $10$ are two natural numbers.

$7 \,=\, 1 \times 7$

The factors of $7$ are two and they are $1$ and $7$. So, the natural number $7$ is a prime number.

The natural number $10$ can be factored in three ways possibly.

$(1) \,\,\,\,\,\,$ $10 \,=\, 1 \times 10$

$(2) \,\,\,\,\,\,$ $10 \,=\, 2 \times 5$

$(3) \,\,\,\,\,\,$ $10 \,=\, 1 \times 2 \times 5$

The factors of $10$ are $1, 2, 5$ and $10$. So, it has four factors. Therefore, the natural number $10$ is not a prime number.

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