# Prime numbers

**Introduction**

A prime number is defined as any natural number with only two factors that are $$1$$ and itself. The number $$2$$ is an example of a prime number. This is because the only way of denoting $$2$$ as a product is $$2 \times 1$$ or $$1 \times 2$$. Also, $$2$$ is the only even prime number.

The first $$10$$ prime numbers are $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, $$17$$, $$19$$, $$23$$, $$29$$.

Some of the important properties related to prime numbers have been stated below.

Every number that is greater than $$1$$ will be divisible by at least one prime number.

Only common factor of any two prime numbers is $$1$$, therefore prime numbers are always co-prime.

A positive integer that is greater than $$2$$ can be expressed in the form of a sum of two primes.

**E1.1: Identify and use prime numbers**

**Composite numbers:**

The numbers that can be obtained by taking the product of two smaller positive integers are known as composite numbers. In other words, a number which has three or more factors is a composite number. For example, $$6$$ is a composite number. We can obtain $$6$$ by multiplying $$6$$ by $$1$$ and also by multiplying $$2$$ by $$3$$.

Therefore, $$6$$ has more than two factors, $$1$$, $$2$$, $$3$$ and $$6$$, so it is not a prime number. Note that the numbers which are not prime, except $$1$$, are composite numbers.

**Identifying Prime Numbers:**

There are two methods to find out if a given number is prime.

Except $$2$$ and $$3$$, every prime number can be denoted in the form of $$6n+1$$ or $$6n-1$$ (not including the multiples of prime numbers, that is, $$2$$, $$3$$, $$5$$, $$7$$, $$11$$) where $$n$$ is any natural number. For example,

$$6(1) – 1 = 5$$

$$6(1) + 1 = 7$$

$$6(2) – 1 = 11$$

$$6(2) + 1 = 13$$

$$6(3) – 1 = 17$$

$$6(3) + 1 = 19$$

$$6(4) – 1 = 23$$

$$6(4) + 1 = 25$$ (Multiple of $$5$$)

We can use the formula, $$n^2+n+41$$ to obtain the prime numbers that are greater than $$40$$. Here, $$n$$ can be a number between $$0$$, $$1$$, $$2$$, $$3$$, …, $$39$$. For example,

$$0^2+0+41 = 41$$

$$1^2+1+41 = 43$$

**Mathematical description of prime numbers:**

Each prime number is a positive number which cannot be expressed as a product of two even integers. There are a number of primes in the number system. Following is the table of all-natural numbers from $$1$$ to $$100$$. The dark boxes represent the prime numbers.

**Worked examples**

**Example 1**: Is $$10$$ a prime number?

**Step 1: Find the factors of $$10$$.**

The factors of $$10$$ are $$1$$, $$2$$, $$5$$ and $$10$$.

**Step 2: write answer according to the definition of prime numbers**

Since, $$10$$ has more than two factors. Therefore, the number $$10$$ is not a prime number.

**Example 2**: Is $$17$$ a prime number?

**Step 1: Find the factors of $$17$$.**

The factors of $$17$$ are $$1$$ and $$17$$.

**Step 2: write answer according to the definition of prime numbers**

Since, the number $$17$$ has only two factors, $$1$$ and $$17$$. Therefore, it is a prime number.

All prime numbers except 2 can be written as a sum of two primes. The only even prime number is 2. Natural number 1 is neither prime nor composite. On dividing a prime number with any other number besides 1 and the number itself the remainder is always not equal to zero. Any two prime numbers are always co prime to each other.