# Introduction to Prime Numbers Notes | Study Quantitative Reasoning for GMAT - GMAT

## GMAT: Introduction to Prime Numbers Notes | Study Quantitative Reasoning for GMAT - GMAT

The document Introduction to Prime Numbers Notes | Study Quantitative Reasoning for GMAT - GMAT is a part of the GMAT Course Quantitative Reasoning for GMAT.
All you need of GMAT at this link: GMAT

What is a Prime Number?

A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself.

• Any number which does not follow this is termed as composite numbers which means that they can be factored into other whole numbers.
• First Ten Prime Numbers: The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Note: It should be noted that 1 is a non-prime number.

Try yourself:Every prime number is

Prime Numbers 1 to 100 List

As we know, the prime numbers are the numbers that have only two factors which are 1 and the number itself. There are a number of primes in the number system. Here is the list provided of prime numbers that are present between 1 and 100, along with their factors and prime factorization.

 Prime Numbers between 1 and 100 Factors Prime Factorisation 2 1, 2 1 x 2 3 1, 3 1 x 3 5 1,5 1 x 5 7 1,7 1 x 7 11 1,11 1 x 11 13 1, 13 1 x 13 17 1, 17 1 x 17 19 1, 19 1 x 19 23 1, 23 1 x 23 29 1, 29 1 x 29 31 1, 31 1 x 31 37 1, 37 1 x 37 41 1, 41 1 x 37 43 1, 43 1 x 43 47 1, 47 1 x 47 53 1, 53 1 x 53 59 1, 59 1 x 59 61 1, 61 1 x 61 67 1, 67 1 x 67 71 1, 71 1 x 71 73 1, 73 1 x 73 79 1, 79 1 x 79 83 1, 83 1 x 83 89 1, 89 1 x 89 97 1, 97 1 x 97

Try yourself:Number of prime numbers from 1 to 50 are

Why 1 is not a prime number?

Conferring to the definition of the prime number which states that a number should have exactly two factors for it to be considered as a prime number. But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered as a Prime number.

Even Prime Number

As we know, the prime numbers are the numbers that have only two factors and the numbers that are evenly divisible by 2 are even numbers. Therefore, 2 is the only prime number that is even and the rest of the prime numbers are odd numbers, hence called odd prime numbers.

Twin Prime Numbers

The prime numbers that have only one composite number between them are called twin prime numbers or twin primes. The other definition of twin prime numbers is the pair of prime numbers that differ by 2 only. For example, 3 and 5 are twin primes because 5 – 3 = 2.

The other examples of twin prime numbers are:

• (5, 7)        [7 – 5 = 2]
• (11, 13)    [13 – 11 = 2]
• (17, 19)    [19 – 17 = 2]
• (29, 31)    [31 – 29 = 2]
• (41, 43)    [43 – 41 = 2]
• (59, 61)    [61 – 59 = 2]
• (71, 73)    [73 – 71 = 2]

Coprime Numbers

The pair of numbers that have only one factor in common between them, are called coprime numbers. Prime factors and coprime numbers are not the same. For example, 6 and 13 are coprime because the common factor between them is 1 only.

Smallest Prime Number

The smallest prime number as defined by modern mathematicians is 2. To be prime, a number must be divisible only by 1 and the number itself which is fulfilled by the number 2.

Largest Prime Number

As of January 2016, the largest known prime number is 2{77,232,917} – 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS).

How to Find Prime Numbers?

The following two methods will help you to find whether the given number is a prime or not.

Method 1:

We know that 2 is the only even prime number. And only two consecutive natural numbers which are prime are 2 and 3. Apart from those, every prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a 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)

Method 2:

To know the prime numbers greater than 40, the below formula can be used.

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

For example:

(0)2 + 0 + 0 = 41

(1)2 + 1 + 41 = 43

(2)2 + 2 + 41 = 47

Example Questions:
Example 1: Is 10 a Prime Number?
Answer – No, because it can be divided evenly by 2 or 5, 2×5=10, as well as by 1 and 10.
Example 2: Is 19 a Prime Number?
Answer – First we need to find the number k, such that k2>19.

Thus the value of k comes out to be 5, as 52 = 25.

The prime numbers less than 5 are 2 and 3.

Clearly, we can see that 19 is not divisible by both 2 and 3.

Therefore, 19 is a prime number.

Note- It should be noted that an integer suppose, P > 1 is termed as a prime number when its only divisors are 1 and P.
Any numeric m > 1 which is not a prime is termed as a composite.

Each composite number can be factored into prime factors, and individually all of these are unique in nature.

Properties of Prime Numbers
Some of the properties of primes are:

• Every number greater than 1 can be divided by at least one prime number.
• Every even positive integer greater than 2 can be expressed as the sum of two primes.
• Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
• Two prime numbers are always coprime to each other.
• Each composite number can be factored into prime factors, and individually all of these are unique in nature.
Prime Numbers and Composite Numbers
 Prime Numbers Composite Numbers A prime number has two factors only. A composite number has more than two factors. It can be divided by 1 and the number itself.For example, 2 is divisible by 1 and 2. It can be divided by all its factors. For example, 6 is divisible by 2,3 and 6. Examples: 2, 3, 7, 11, 109, 113, 181, 191, etc. Examples: 4, 8, 10, 15, 85, 114, 184, etc.
The document Introduction to Prime Numbers Notes | Study Quantitative Reasoning for GMAT - GMAT is a part of the GMAT Course Quantitative Reasoning for GMAT.
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