Prime Numbers – Know All About them

Prime Numbers are basically positive integers that have just two factors: one and the integer itself. For example, factors of 10 are 1, 2, 5, and 10, for a total of four factors. We may alternatively define prime numbers as numbers that are only divisible by 1 and the number itself, or it may also be deduced as a positive number or integer that is not a product of any other two positive integers. One must keep in mind that 1 is neither prime nor composite. As a result, all prime numbers are bigger than one. We may also state that, with the exception of 1, the remaining numbers are classed as prime and composite numbers.

What Are Prime Numbers?

A Prime Numbers is defined as a positive integer with exactly two factors. If p is a prime, its only factors must be 1 and p itself. Any number that does not fall into this category is said to as a composite number, which signifies that it may be factored into other positive integers.

How To Find Prime Numbers?

According to the definition of Prime Numbers, these have just two factors. The original number and 1 would be the two components. As a result, we must determine the integers that have only two factors. This is feasible thanks to a simple procedure known as prime factorisation.

It is simple to find the primes for lower numbers, but for greater numbers, we must devise a new method. As a result, we have demonstrated how to assess prime numbers not only for smaller digits but also for larger integers.

Steps Involved To Find Prime Numbers Via Factorization Method

Factorisation is the most effective method for discovering prime numbers. The following stages are involved in adopting the factorisation method:

Step 1: To begin, determine the factors of the given integer.

Step 2: Determine the number of factors that make up that number.

Step 3: If there are more than two factors, the number is not a prime number.

Demonstration To Find Prime Numbers Via Factorization Method

Let the number be 21

Now, 21 can be written as 7 × 3. So, the factors of 21 here are 1, 3, 7  and 21. Since the number of factors of 21 is more than 2, it is not a prime number but a composite number.

Let the number be 36

Now, 36 can be written as 2 × 3 × 2 × 3.  So, the factors of 36 here are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Since the number of factors of 36 is more than 2, it is not a prime number but a composite number.

Let the number be 13

Now, 13 can be written as 1 x 13.  So, the factors of 13 here are 1 and 13. Since the number of factors of 13 is not more than 2, it is a Prime Number

How To Know Whether A Large Integer Is Prime Or Not

There are various Prime Number Formulas that may be used to determine the primes. Follow the procedures below to determine if a big number is a prime number or not:

Step 1: Look at the number’s units location. It is not a prime number if it ends with 0, 2, 4, 6, or 8.

“Numbers that finish in 0, 2, 4, 6, and 8 are never prime numbers since they would always have 2 as their factor.”

Step 2: Add the digits of that number together. If the total is divisible by three, the integer is not prime.

“Numbers whose sum of digits is divisible by three are never prime numbers,” says the rule.

Step 3: Once the falsehood of steps 1 and 2 have been established, find the square root of the provided integer.

Step 4: Divide the provided integer by the prime numbers that are less than its square root value.

Step 5: If the number is divided by any of the prime numbers less than its square root, it is not prime; otherwise, it is.

The exception is that if a huge number ends in 5, it is always divisible by 5. As a result, it is not a prime number.

Illustration For Better Understanding

Illustration 1

• Let the number be 2168376
• Since the unit digit of 2168376 is 6, it is not a prime number.

Illustration 2

• Take a number, say, 26577
• The unit digit of this number is not 0, 2, 4, 6 or 8
• Now, take the sum of digits which will be: 2 6 5 7 7 = 27
• Since 27 is divisible by 3, 26577 is not a prime number.

Illustration 3

• Take another number, say, 2345
• Since the number ends with 5, therefore, it is divisible by 5.
• 2345/5 = 469
• Hence, apart from 1 and 2345, 5 is also a factor.
• Therefore, 2345 is not a prime number

Prime Numbers Between 1 And 200

Here is a collection of prime numbers ranging from 1 to 200, from which we may learn and cross-check to see if they have any other factors.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Important Point About Prime Numbers

• The only even prime number is ‘2’; all other primes are odd numbers.
• The only two consecutive prime numbers are ‘2’ and ‘3’.
• The sum of two prime numbers may be used to represent any even integer bigger than 2.
• According to current mathematics, the smallest prime number is 2. A number must be prime if it is divisible only by 1 and the number itself, which is satisfied by the number 2. 
• Coprime numbers are pairs of numbers that have just one element in common. The terms prime factors and coprime numbers are not interchangeable. For example, 6 and 13 are coprime since their single shared element is 1.

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