Prime Numbers and Composite Numbers Explained

It is crucial to understand the difference between prime and composite numbers in Mathematics. While composite numbers are numbers with more than two factors. They are just the opposite of prime numbers. Prime numbers are those that have just two factors, i.e. 1 and the number itself. A prime number is always a natural number. All the natural numbers that are not prime numbers fall into the category of composite numbers. So, composite numbers are divisible by more than two numbers. In this article, we will learn the following:

• What are prime and composite numbers?
• Difference between prime and composite numbers
• Properties of prime and composite numbers
• How to identify prime and composite numbers
• Facts about prime and composite numbers
• Composite numbers list
• Prime and composite numbers chart

What are Prime Numbers? 

A prime number is a positive integer. It has exactly two factors, 1 and the number itself. So, if n is a prime number, its factors will be 1 and n itself. We can also define a prime number as a number that is a positive integer and is not a product of any other two positive integers other than the number itself and 1. 

Interesting Prime Numbers

There are some interesting facts about prime numbers. Some of them are listed here.

Prime Numbers

Smallest prime number is 2

Largest Prime Number 2^(82,589,933) – 1 is the recent largest prime number. Mathematicians are still finding more.

Twin Primes The prime numbers whose difference is two are twin prime numbers. For example,  3 and 5, 5 and 7, 11 and 13 are sets of twin prime numbers. In other words, they are two consecutive prime numbers that have only one number between them.

Co-Prime Numbers Co-prime numbers are the two numbers that have only 1 as a common factor. For example, 2 and 3, 4 and 5, 3 and 7, 4 and 9 are co-prime numbers.

Properties of Prime Numbers

The following are some properties of prime numbers: 

• Every prime number is greater than 1
• Each prime number is a factor of itself
• Every number that is greater than 1 can be divided by at least one prime number.
• Every positive integer that is even and greater than 2 can be written as the sum of two prime numbers.
• 2 is the only even prime number. 
• All prime numbers are odd except 2.
• Two prime numbers are coprime to one another.
• Every composite number can be factorized into prime factors.

Prime Numbers List

Here is a list of prime numbers from 1 to 100. Students can understand the concept of prime numbers from this list and create a list beyond 100 for practice.

Prime Numbers List

2, 3, 5, 7, 1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

What are Composite Numbers?

Unlike a prime number, a composite number is a number that has more than two factors.

We can define composite numbers as the numbers that can be generated on multiplication of the two smallest positive integers and contain at least one more divisor in addition to the number ‘1’ and itself. All even numbers that are greater than 2 are amongst the many examples of composite numbers. 

So, are all even numbers composite? 

No, 2 is an even prime number. In fact, it is the only even number that is prime. Therefore, we cannot say that all even numbers are composite numbers.

Types of Composite Numbers

Composite numbers are of two types:

• Odd Composite Numbers 
• Even Composite Numbers 

Odd Composite Numbers

All odd numbers greater than 1  that are not prime are odd composite numbers. 

The examples of odd composite numbers are 9, 15, 21, and more.

Even Composite Numbers

Even composite numbers include all even integers that are not prime numbers.
The examples of even composite numbers are 4, 6, 8, 10, and more.

Properties of Composite Numbers

The properties of composite numbers are as follows:

• Every composite number has more than two factors
• The factors evenly divide composite numbers
• Composite numbers are their own factors too.
• 4 is the smallest composite number
• Each composite number has at least two prime numbers as its factors.
• A composite number is divisible by other composite numbers too.

Composite Numbers List

Here is a list of all composite numbers from 1 to 100. Students can understand the concept of composite numbers from this list and create a list beyond 100 for practice. 

Composite Numbers List

4. 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

How to Find Composite Numbers?

You must find the factors of a number to determine if it is a composite number. If there are more than two factors, then the number is a composite number. We can perform the divisibility test to find whether a number is prime or composite. 

In the divisibility test, we divide the number by common factors such as 2, 3, 5, 7, 11, and 13. If these factors fail to divide the number completely, then the number is a prime number. For example, 22 is divisible by 2, which means it has a factor 2 other than 1 and 22. Therefore, 22 is a composite number.

How to Find Prime Numbers?

There are two methods that help determine if given numbers are prime or composite numbers. 

Method 1:

Apart from 2 and 3, you can write every prime number in the form of 6n + 1 or 6n – 1, where n is a natural number.

For example:

6(1) – 1 = 5

6(1) + 1 = 7

Method 2:

For determining a number more than 40 as a prime number, we can use the following formula: 

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

For example:

(0)2 + 0 + 41 = 41

(1)2 + 1 + 41 = 43

(2)2 + 2 + 41 = 47

Difference between Prime and Composite Numbers

There are multiple differences between prime numbers and composite numbers. The following tables enumerate some key differences between the two.

Prime Numbers	Composite Numbers
They have 2 factors. One and the number itself.	They have more than two factors.
It can be written in the form of the product of two numbers.	They can be written as the product of two or more numbers
Example: The factors of 7 are 1 and 7.	Example: The factors of 6 are 1, 2, 3, and 6.

Prime and Composite Numbers Chart

Earlier mathematicians used numerical tables to record all of the primes or prime factorizations. The sieve of Eratosthenes presented a list of prime numbers. This method provides a chart- Eratosthenes chart. The following chart shows prime numbers up to 100 in colored boxes. All numbers other than the colored boxes are composite numbers.

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100