Finding prime numbers is not a mammoth task. All that it requires is a simple task of understanding what a prime number is. Here, let us learn what a prime number is and then move on to its history, giving an insight into how various theorems were proposed on prime numbers. It is equally important to learn its properties and also about co-prime numbers. Finally, we can learn how to identify a prime number using various methods.

*Prime Number: *

A prime number is a number that could most effective be divided via itself and one without remainders. Right here, we explain what exactly this means.

Prime numbers are the ones that have the handiest two elements, that are, one and the number that is taken into account. Take into account the example of number seven, which has the best factors one and seven. It indicates it is a prime number. Let’s take some other examples of the number 9, which has three elements, i.e. one, three, and nine. This means 9 is not a prime number. If we take the example of number one, we know that it has just one factor. So, it is not a prime number as a prime number ought to have precisely two factors. This indicates one is not a prime and not a composite number. It is a unique number.

*History of Prime numbers:*

Euclid put forth the idea of prime numbers. Prime numbers have piqued human interest since time immemorial. Even nowadays, mathematicians are looking for prime numbers with special properties.

The Numbers shaded in yellow in the above table are prime numbers. Did you notice all of the prime numbers in the table? Did you try to see if each number is divisible via the smaller numbers? Then, you invested lots of effort and time. Eratosthenes changed into one of the finest mathematicians. They lived many years after Euclid, came up with a wonderful idea to determine all of the prime numbers as much as a given number. Assume you need to find the prime numbers as much as n. We can generate the listing of all numbers from 2 to n. Beginning from the smallest prime number two, we will score out all the multiples of 2, besides 2, from the list. Similarly, assign the next value of p that’s a prime number extra than 2.

*Properties of a Prime number:*

- A prime number is a variety of more than 1 with the most effective factors – themselves and one.
- Prime numbers need to have exactly two factors.
- A prime number cannot be divided by using other numbers without leaving a remainder.
- An example of a prime number is thirteen. It can handiest be divided by way of one and thirteen. Dividing a prime number by another number consequences in numbers leftover, e.g., 13 ÷ 6 = 2 remainder one.
- 15 is not an instance of a prime number because it could be divided via 5 and three as well as with the aid of itself and one.
- 15 is an instance of a composite number because it has more than two elements.

*Prime and Co-Prime numbers:*

There’s a difference between prime numbers and co-prime numbers. If a couple of numbers has nothing in common apart from one, they are known as co-prime numbers. Co-prime numbers are usually considered in pairs, whilst a single number can be interpreted as a prime number. Co-prime numbers can be prime, but they can also be composite. The best criteria to be met is that the Greatest Common Factor of co-prime numbers is constantly one.

**FINDING PRIME NUMBERS USING FACTORIZATION**

The most common method used to find prime numbers is by means of the factorization technique.

The steps concerned in identifying prime numbers by use of the factorization technique is:

- First, find the factors of the given number (factors are the number that can divide the given number without having any numbers left)
- Then look for how many factors that number has.
- As a result, If the entire number of factors exceeds two, it isn’t always a prime number but a composite number.

**For Example: Is 46 a prime number?**

Factors of 46= 1, 2, 23, 46

Since the factors of 46 exceed two. It is not a prime number but a composite number.

Now, if we take the number 13, the Prime factorization of 13 is 1,13. There are only two factors. So, it is a Prime number.

Prime numbers can also be found by the other strategies using the general formula. *The techniques to discover prime numbers are:*

**Method 1:**

Continuous numbers, which are natural numbers and prime numbers, are 2 and 3. Apart from 2 and 3, each prime number can be written within 6n 1 or 6n – 1, where n is a natural number.

E.g., 6(1) + 1= 7

6(1) – 1= 5

6(2) – 1 = 11

6(2) + 1 = 13

6(3) – 1 = 17

6(3) + 1 = 19

6(4) + 1= 25

6(4) – 1= 23…… And so on.

**Method 2:**

To find the prime numbers greater than 40, the general formula that can be used is n2 n 41, where n are natural numbers 0, 1, 2, ….., 39

For example:

(0)2 + 0 + 0 = 41

(1)2 + 1 + 41 = 43

(2)2 + 2 + 41 = 47

+ 3 + 41 = 53

(4)2 + 2 + 41 = 59…..so on

**Example 1: Why is 15 not a prime number?**

**Solution:**

The factors of 15 are 1, 3, 5, 15. Thus, 15 has four factors. Since the number of factors of 15 is four numbers, it is NOT a prime number (15 is a composite number).

**Example 2: Is 1423 a prime number or a composite number?**

**Solution: **

We need to bring out its factors first to identify if 1423 is a prime number or a composite. The factors of 1423 are 1, 1423. Thus, 1423 does not have anything apart from the two factors. Since the number of factors of 1423 is only two numbers, it is a prime number.

**Example 3: Is 17 a Prime Number or not?**

**Solution:**

We can check if the number is prime or not in two ways.

**Method 1:**

The formula for the prime number is 6n – 1

Let us write the given number in the form of 6n – 1.

6(3) + 1 = 18 – 1 = 17

**Method 2: **

Check for the factors of 17

17 has only two factors, 1 and 17.

Therefore, by both methods, we get 17 as a prime number.

**Example 4: Is 61 a prime number or not?**

**Solution:**

**Method 1:**

To identify the prime numbers greater than 40, the below method can be used.

n2 + n + 41, where n = 0, 1, 2, ….., 39

Put n= 4

42 + 4 + 41 = 16 + 4 + 41 = 61

**Method 2:**

61 has only factors 1 and 61.

So, 61 is a prime number by both methods.

**Conclusion **

Numbers are what mathematics is about. And prime numbers are one of the several kinds of numbers. It is a fundamental concept that will help you to solve your sums faster. With much practice, you can easily memorise all the prime numbers from 1 to 100 with ease.

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