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

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.

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.

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

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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

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.

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We cannot determine the largest prime number. 

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

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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.

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).