Prime numbers are the category of natural numbers which are greater than 1 and are divisible only by 1 and itself. Prime numbers do not give zero as remainder when divided by any other number than 1 or itself. The property of a number being prime is known as **primality** and the method of testing a number whether it is prime, is called **primality test**.

According to Primality test, following steps should be followed while testing a number for being prime (let us consider a natural number x which is to be tested):

- Divide x by 2. If the quotient is integer which means x is divisible by 2, then it is concluded that the number is not prime, else follow next step.
- Divide x by 3. If the quotient is integer which implies that 3 is a divisor of x. In such case it is concluded that the given number is not prime, otherwise follow next step.
- Continue dividing by successive numbers and check for the remainder.
- Stop the process of division till dividing by $\sqrt{x}$, because after $\sqrt{x}$, the factors start repeating themselves.
- If from 2 to $\sqrt{x}$, the number x is not divisible by any number, then it means that x is prime, otherwise not prime.

No need to check by numbers like 2, 6, 9, 10, 12 etc, if already checked for their prime factors 2, 3 and 5 etc.Let us take an example to understand this method clearly.

**Example:** Check whether 91 is prime.**Solution:** **Step 1 -** On dividing 91 by 2, we get 45.5 (quotient is not integer). Therefore, 91 is not divisible by 2.

Step 4 -

Hence, 91 is not a prime number.

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