Prime Numbers
Prime numbers are the numbers that have only two factors, that are, 1 and the number itself. Consider an example of number 5, which has only two factors - 1 and 5. This means it is a prime number. Let's take another example of the number 6, which has more than two factors- 1, 2, 3, and 6. This means 6 is not a prime number. Now, if we take the example of the number 1, we know that it has only one factor. So, it cannot be a prime number as a prime number should have exactly two factors. This means 1 is neither a prime nor a composite number, it is a unique number.
Before we begin, let us watch this video that will give us a little teaser before we explore the mystery of prime numbers.
What are Prime Numbers?
A number greater than 1, with exactly two factors, 1 and itself, is defined as a prime number in mathematics. In other words, if a number cannot be divided into equal groups, then it is a prime number. 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. For example, 7 cannot be divided into groups of equal numbers.
Definition: Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number.
History of Prime Numbers
Prime numbers created human curiosity since ancient times. Even today, mathematicians are trying to find prime numbers with mystical properties. Euclid proposed the theorem on prime numbers - There are infinitely many prime numbers.
Do you know all the prime numbers from 1 to 100? Did you check if each number is divisible by the smaller numbers? Then, you definitely invested a lot of time and effort. Eratosthenes was one of the greatest scientists, who lived a few decades after Euclid, designed a smart way to determine all the prime numbers up to a given number. This method is called the Sieve of Eratosthenes. Suppose you have to find the prime numbers up to n, we will generate the list of all numbers from 2 to n. Starting from the smallest prime number p = 2, we will strike off all the multiples of 2, except 2 from the list. Similarly, assign the next value of p which is a prime number greater than 2. Repeat this process till p ≤ ⎷n
List of Prime Numbers from 1 to 100
There are 25 prime numbers from 1 to 100. Here is the list of prime numbers from 1 to 100:
| List of Numbers | Prime Numbers |
| Between 1 and 10 | 2, 3, 5, 7 |
| Between 11 and 20 | 11, 13, 17, 19 |
| Between 21 and 30 | 23, 29 |
| Between 31 and 40 | 31, 37 |
| Between 41 and 50 | 41, 43, 47 |
| Between 51 and 100 |
53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
Properties of Prime Numbers
Prime numbers have many properties. Some of the important properties of prime numbers are given below:
- A prime number is a whole number greater than 1.
- It has exactly two factors, that is, 1 and the number itself.
- There is only one even prime number, that is, 2.
- Any two prime numbers are always co-prime to each other.
- Every number can be expressed as the product of prime numbers.
Prime Numbers vs Coprime Numbers
Prime Numbers and Composite Numbers
A prime number is a number greater than 1 that has exactly two factors.
Example:
5 can be factorized in only one way, that is, 1 × 5 (OR) 5 × 1. It has only two factors, which are, 1 and 5. Therefore, 5 is a prime number.
A composite number is a 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.
Prime Numbers and 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:
- 5 and 9 are co-primes.
- 6 and 11 are co-primes.
- 18 and 35 are co-primes.
Co-prime numbers need not necessarily be prime numbers.
Easy Way to Find Prime Numbers
There are different ways to find prime numbers. Let us go through two of these methods.
Method 1: Substitute whole numbers for n in the formula 'n2 + n + 41'. This formula will give you all the prime numbers greater than 40. Let's substitute a few whole numbers and check.
02 + 0 + 41 = 0 + 41 = 41
12 + 1 + 41 = 2 + 41 = 43
22 + 2 + 41 = 6 + 41 = 47
Continuing like this, you can calculate all the prime numbers greater than 40.
Method 2: Every prime number, apart from 2 and 3, can be written in the form of '6n + 1 or 6n - 1'. So, if you have any number different from 2 and 3, you can check if it is prime or not by trying to express in the form of 6n + 1 or 6n - 1
6(1) - 1 = 5
6(1) + 1 = 7
6(2) - 1 = 11
6(2) + 1 = 13
Now, we know that the numbers 5, 7, 11, and 13 are prime.
List of Odd Prime Numbers
A prime number chart is a chart that shows the list of prime numbers in a systematic order. Given below is the prime number chart from numbers 1 to 100 that shows the list of odd prime numbers(highlighted in yellow)
Is There A Pattern In Prime Numbers?
The set of prime numbers between any two numbers can be found by following a pattern. The following figure shows a few prime numbers encircled and striking off all the numbers divisible by these prime numbers. You can follow this pattern until you reach the square root of the larger number, that is, 100
Solved Examples
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Example 1: Which of the two numbers is a prime number, 13 or 15?
Solution
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. Therefore, 13 is a prime number.
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Example 2: Why is 20 not a prime number?
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). Therefore, 20 is not a prime number.
FAQs on Prime Numbers
What are the Examples of Prime Numbers?
A number greater than 1, with exactly two factors, 1 and itself, is defined as a prime number in mathematics. Here are some examples of prime numbers: 2, 3, 31, 101, 149, etc.
What is the Easiest Way to Find a Prime Number?
One of the easiest methods to find that a given number p, is a prime number, is to check the number of factors of the number p. If p has exactly 2 factors, 1 and p, then we say that p is a prime number.
Why is 2 a Prime Number?
The factors of 2 are 1 and 2. Since 2 has exactly two factors, therefore it is a prime number. Interestingly, 2 is the only prime number.
What are all the 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
What is the Formula to Find Prime Numbers?
Suppose you have a number p and n is the smallest natural number such that n2 ≥ p. If p is divisible by any number less than or equal to n, then p is not prime otherwise, p is prime.
Which is the Smallest Prime Number?
The smallest number which has exactly two factors 1 and itself is 2. Therefore, the smallest prime number is 2.
What is the Difference Between a Prime and a co-prime Number?
A prime number is a number that has only two factors, that is, 1 and the number itself. For example, 2, 3, 5, 7 are prime numbers.
Co-prime numbers are the numbers whose highest common factor is 1. For example, 2 and 3 are co-prime numbers.
Can Prime Numbers be Negative?
The prime numbers should be only whole numbers, and all the whole numbers are greater than 1. Therefore, a prime number can't be negative.
Which is the Largest known Prime Number?
The largest prime number is 282,589,933 - 1, which is having 24,862,048 digits.