What is a Prime Number?

In this article we have covered the concept of Prime Number in Mathematics. Learn what is a prime number, what are the different prime numbers between 1-100 and 101-1000, and how to determine if a number is a prime number.

Prime numbers are one of the building blocks of mathematics. Prime numbers are relatively basic and easy to understand with a minimum understanding of mathematics. In this article, we will explore what is a prime number.

Content

• What are Prime Numbers?
• Table of Prime Numbers Up To 100
• Prime Numbers 101 -1000
• Is 1 Prime?
• Factors
• Factorization of Numbers
• How To Know If A Number Is Prime?

What is a Prime Number?

Prime numbers are subsets of natural numbers. They have only two divisors or factors: 1 and the number itself. A number that can be divided by one is a prime number.

There are 25 prime numbers between 1 and 100, they 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.

The property for numbers to be prime is known as primality. Note that all the only prime numbers are odd, except for 2. This is quite obvious since, from 4, all of them can be divided, at least, by 2 because they are even.

Table of Prime Numbers Up To 100

In the below table, the highlighted ones are prime numbers.

Prime Numbers 101 -1000

Is 1 a Prime Number?

Currently, the mathematical community does not consider 1 in the list of prime numbers. This was already defined by a convention emphasizing that prime numbers only have two factors, the digit itself and 1. 

1 is not a prime number, as it has a single factor, 1.

However, until the 19th century, mathematicians mainly considered it a prime. Many mathematical works remain valid despite considering 1 as a prime, like several huge published lists of prime numbers beginning with 1.

Factors 

Prime numbers help find the factors of any natural number. A factor is a number by which a natural number can be divided. Think of this as “is divisible by”. 

Example:

The factors of 14 are 14, 7, 2, 1, since 2 x 7 = 14, and 1 x 14 = 14.

Factorization of Numbers

Prime numbers have many applications in advanced mathematics, primarily when used as factors of other numbers. Non-prime numbers can be divided into prime factors. 

For example, 12 in 4 x 3 can be divided into prime factors like this, 2 x 2 x 3.

How To Know If A Number Is Prime?

To determine if a number is prime, divide it by all the prime numbers less than it. If an inexact division is reached in which the quotient is equal to or less than the divisor, the number is prime.

We can use divisibility rules to determine if a number is prime. When any digit is divisible by other than 1 and itself, it is no longer prime. 

So let’s understand this using one of the easy methods to check if the number is a prime. This method is called the Divisibility Test.

Divisibility Test by 2

All even numbers, those ending in 2, 4, 6, 8, and 0, are divisible by 2. In this case, the numbers divisible by 2 are not prime.

Divisibility Test by 3

Any number whose sum of its digits is equal to 3 or a multiple of 3 is divisible by 3. Therefore, it is not prime.

Divisibility Test by 5

Any number that ends in 5 or 0 is divisible by 5 and is not prime. Example:

Divisibility Test by 7

To find out if a number is divisible by 7, we follow these steps:

1. Separate the first digit from the right

2. Multiply this number by 2

3. Subtract this product from the remaining number on the left

4. Repeat steps 1 to 3 until we reach a multiple of 7 or zero.

The result shows that the number is divisible by 7. Hence it is not a prime number.

Divisibility Test by 11

To find out if a number is divisible by 11, we follow these steps:

1. Add the digits in even places 
2. Add the digits in odd places
3. Subtract the first sum from the second.
4. If the result is zero or a multiple of 11, the number is divisible by 11.

The result shows that these numbers, 165 and 2695, are divisible by 11; hence they are not prime numbers.

You May Like – Introduction to Probability

Difference Between Prime Numbers and Composite Numbers

In Mathematics, the division between prime and composite numbers is a fundamental factor. 

As discussed, prime numbers are only divisible by themselves, and 1. For example, 7 is a prime number because it is only divisible by 1 and 7.

Another example, 13 is a prime number. As in the previous case, it is only divisible by 1 and 13. 

On the contrary, the composite numbers are those natural numbers that have some divisor apart from themselves and 1 and, therefore, can be factored. The number 1, by convention, is considered neither prime nor composite.

25 is a composite number divisible by 1, 5, and 25. So is 14 because it is divisible by 1, 2, 7, and 14.

To sum it up –

Prime Numbers	Composite Numbers
A prime number has two factors – 1 and the number itself.	A composite number has two or more factors.
All prime numbers are Odd, except 2.	All numbers except prime numbers and 1 are composite numbers.
Examples – 2, 5, 7, 11, 13, 17, 19, 23, 29	Example – 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20

Also Read – Measures of Central Tendency: Mean, Median and Mode

Smallest Prime Number

2 is the smallest prime number. The rest of the even numbers are divisible by themselves, 1 and 2 at least, meaning they will have at least three factors.

Largest Known Prime Number

The most significant known prime number is the result of raising 82,589,933 to power 2 and subtracting 1 from it.

Patrick Laroche calculated it on December 7, 2018, through the GIMPS (Great Internet Mersenne Prime Search) network. This number has 24,862,048 digits.

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