# Prime Numbers

Here's a little secret. Number 2 is always alone because it is the only even prime number in existence. But what is a prime number, we hear you ask. Well, in this chapter, you will learn about the prime number definition, prime number examples, how to find and identify prime numbers, the prime number table, and prime numbers from 1 to 100. Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions on Prime Number at the end of the page.

Before we begin, let us watch this video that will give us a little teaser before we explore the mystery that is prime numbers.

## Lesson Plan

 1 What is a Prime Number? 2 Tips and Tricks 3 Solved Examples on Prime Numbers 4 Important Notes on Prime Numbers 5 Interactive Questions on Prime Numbers

## What is a Prime Number?

If a number can be divided into equal groups, then it is not a prime number. It is a Composite number.

What is a prime number?

If a number cannot be divided into equal groups, then it is a prime number.

Keep in mind:

• group's size cannot be 1.
• groups cannot have fractional quantities like $$\frac{1}{2}$$ in each group.

Do you think that you can divide things into groups with an equal number of things?

For example, we can divide 12 circles into groups with an equal number of circles in different ways:  This is because 12 can be factorized in the following ways:

• $$12 = 4 \times 3$$
• $$12 = 6 \times 2$$

We know that 12 can be factorized in the following ways as well:

• 3 groups of 4 because $$3 \times 4 =12$$
• 2 groups of 6 because $$2 \times 6 =12$$

(1 group of 12 or 12 groups of 1 don't count in our current system)

Quick recall: 1, 2, 3, 4, 6, and 12 are called factors of 12.

We understand from the above example is that we can divide a number into groups with equal numbers of items/elements only if it can be factorized as a product of two numbers.
But it is not possible to divide all numbers into groups with equal numbers.
For example, 7 cannot be divided into groups of equal numbers. This is because 7 can only be factorized as follows:

• $$7 \times 1 = 7$$
• $$1 \times 7 = 7$$

Quick recall: 1 and 7 are the only factors of 7

In the above examples:

• 12 is NOT a prime number (it means it is a composite number) because it could be divided into groups of equal numbers).
• 7 is a prime number because it could NOT be divided into groups of equal numbers.

### Definition of Prime Numbers

Any whole number greater than 1 that has exactly two factors, 1 and itself, is defined as a prime number in mathematics.

Another equivalent definition is:

Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number in mathematics.

Any number greater than 1 that is not a prime number, is defined to be a composite number. Thus composite numbers will always have more than 2 factors.

Any whole number greater than 1 is either a prime or composite number.

For example:

• The number 15 has more than two factors: 1, 3, 5, and 15
Hence it is a composite number.

• On the other hand, 13 has just two factors: 1 and 13
Hence it is a prime number.

### Examples of Prime Numbers

Here are some other examples of prime numbers:

2, 3, 31, 101, 149, etc.

You can find more examples of prime numbers under "List of Prime Numbers from 1 to 100" and "List of Prime Numbers from 101 to 200" sections of this page.

### Is 1 a Prime Number?

• 1 is neither prime nor composite, as the definition of prime numbers talks only about the whole numbers that are greater than 1
• 1 has only one factor. So it cannot be a prime number as a prime number should have exactly two factors.
• 1 has only one factor. So it cannot be a composite number as a composite number should have more than two factors.

### Properties of Prime Numbers

The properties of prime numbers are:

• A prime number is a whole number greater than 1
• A prime number has exactly two factors, i.e. 1 and itself.

### Prime and Composite Numbers

A prime number is a whole number greater than 1 that has EXACTLY two factors.

Example:

5 can be factorized in only one way, i.e. $$1 \times 5$$ (OR) $$5 \times 1$$ It has only two factors 1 and 5. Therefore, 5 is a prime number.

A composite number is a whole number greater than 1 that has MORE THAN two factors.

Example:

4 can be factorized in multiple ways.  So the factors of 4 are 1, 2, and 4. It has more than two factors.

Therefore, 4 is a composite number.

Are you excited to know what are twin prime numbers and what are co-prime numbers?

Here you can see.

### Twin Prime Numbers

Twin prime numbers are two prime numbers that have only one composite number in between.

In other words, twin prime numbers are two consecutive prime numbers.

Examples of twin prime numbers:

1. 3 and 5 are twin primes.
2. 5 and 7 are twin primes.
3. 11 and 13 is another pair of twin primes.

### Co-Prime Numbers

If a pair of numbers has no common factor apart from 1, then the numbers are called co-prime numbers.

Examples of co-prime numbers:

1. 5 and 9 are co-primes.
2. 6 and 11 are co-primes.
3. 18 and 35 are co-primes.

Co-prime numbers don't actually need to be prime numbers.

## List of Prime Numbers from 1 to 100

There are 25 prime numbers from 1 to 100

The prime numbers from 1 to 100 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

Here is the prime numbers table with the list of prime numbers from 1 to 100:

 Prime numbers between 1 and 10 2, 3, 5, 7 Prime numbers between 11 and 20 11, 13, 17, 19 Prime numbers between 21 and 30 23, 29 Prime numbers between 31 and 40 31, 37 Prime numbers between 41 and 50 41, 43, 47 Prime numbers between 51 and 100 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

## List of Prime Numbers from 101 to 200

There are 21 prime numbers from 101 to 200.

The prime numbers from 101 to 200 are:

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199

Here is the prime numbers table with the list of prime numbers from 101 to 200:

 Prime numbers between 101 to 110 101, 103, 107, 109 Prime numbers between 111 to 120 113 Prime numbers between 121 to 130 127 Prime numbers between 131 to 140 131, 137, 139 Prime numbers between 141 to 150 149 Prime numbers between 151 to 200 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

## Prime Numbers Chart

Here is the prime numbers chart from 1 to 100 Here is how we can find the list of prime numbers between 1 and 100 using the sieve of Eratosthenes algorithm.

We just strike off 1 as it is NOT a prime number. Then we strike off all the multiples of other numbers.

Then, the remaining numbers are the prime numbers.

## Largest Prime Number

We cannot determine the largest prime number.

The largest known prime number is $$2^{82,589,933}$$-1 having 24,862,048 digits.

## Smallest Prime Number

The smallest prime number is 2

2 is a prime number as it has exactly two factors (which are 1 and 2)

In fact, 2 is the only even prime number.

## How to find Prime Numbers

If you would like to know whether any given number is prime, follow these steps:

• Write all the factors of the given number.
• Count the factors.
• If the number of factors is EXACTLY 2 then the given number is prime.

If the number of factors is more than 2 then the given number is NOT prime (in that case, it is called a composite number). Tips and Tricks
1. Any even number other than 2 is NOT a prime number.
2. 2 is the only even prime number.
3. Any two-digit number that ends with 0 or 5 is NOT a prime number as any such number is divisible by 5.
4. If the sum of the digits of a number (other than 3) is divisible by 3 then the number is NOT a prime number.

## Solved Examples

 Example 1

Is 6 prime or NOT?

Solution

The given number is 6.

The factors of 6 are  1, 2, 3, and 6.

Thus 6 has 4 factors.

Since the number of factors of 6 is NOT exactly 2, it is NOT a prime number (so 6 is a composite number).

 ∴ 6 is NOT a prime number.
 Example 2

Is 37 prime or NOT?

Solution

The given number is 37.

The factors of 37 are 1 and 37.

Thus 37 has exactly two factors.

Since the number of factors of 37 is exactly 2, it is a prime number.

 ∴ 37 is a prime number.
 Example 3

Is 20 prime or NOT?

Solution

The given number is 20.

The factors of 20 are 1, 2, 4, 5, 10, and 20.

Thus 20 has more than two factors.

Since the number of factors of 20 is NOT exactly 2, it is NOT a prime number (20 is a composite number).

 ∴ 20 is NOT a prime number.

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