Prime number are positive natural number greater than 1 (N > 1) that has only two divisors i.e. 1 and itself. In other words, a prime number is a number that cannot be formed by multiplying two smaller natural numbers (except 1 and itself). For example, 2, 3, 5, 7, 11, and 13 are prime numbers.
What are Prime Numbers?
A prime numbers are a natural number that are divisible by 1 or itself. In other words, Prime Number is a positive integer greater then 1 and divisible by 1 and itself. Some examples of Prime Numbers are: 2, 3, 5, 7, 11, … etc.
Prime Number are Natural Numbers greater than 1 and divisible by 1 or itself. The smallest prime number is 2, and the smallest odd prime number is 3. The only even prime number is 2.
What is the Meaning of Prime Number?
A prime number is a natural number greater than 1 that has only 1 divisors (1 and the number itself). Prime number has only two factors i and the number itself. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers. The number 1 is not considered a prime number because it has only one divisor. Prime numbers have many applications in various fields including – cryptography, number theory, computing, … etc.
Prime Number Factors
The Prime Number has only two factors i.e. 1 and the prime number itself. This is because a prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, the factors of the prime number 7 are 1 and 7, and the factors of the prime number 13 are 1 and 13.
Prime Numbers in Given Range
First 10 Prime Numbers
The First 10 Prime Numbers are – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Prime Numbers in Given Range [1 to 1000]
Prime Numbers in Range | Prime Numbers |
---|---|
Range [1, 100] | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
Range [100, 200] | 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 |
Range [200, 300] | 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293 |
Range [300, 400] | 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397 |
Range [400, 500] | 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499 |
Range [500, 600] | 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599 |
Range [600, 700] | 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691 |
Range [700, 800] | 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797 |
Range [800, 900] | 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887 |
Range [900, 1000] | 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 |
Prime Numbers 1 to 100
Here is a list of prime numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Prime Numbers 1 to 200
Here is a list of prime numbers from 1 to 200:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
Prime Numbers 1 to 1000
Here is a list of prime numbers from 1 to 1000:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997
Prime Numbers Chart
History of Prime Numbers
Eratosthenes, a Greek mathematician (275-194 B.C.), is credited with the discovery of prime numbers. He devised a method akin to using a sieve to sift out prime numbers from a list of natural numbers while filtering out composite numbers.
To practice this method, students can write down the positive integers from 1 to 100, circle the prime numbers, and mark the composites with a cross. This activity is a practical application of the Sieve of Eratosthenes.
Properties of Prime Numbers
Prime numbers have several interesting properties that make them fundamental in mathematics. Some properties of prime numbers are:
- Divisibility: A prime number is only divisible by 1 and itself.
- Unique Factorization: Every natural number greater than 1 can be uniquely expressed as a product of prime numbers. This is known as the fundamental theorem of arithmetic.
- Density: Prime numbers become less frequent as numbers get larger, but they still appear infinitely often. This is known as the infinitude of prime numbers.
- Distribution: Prime numbers do not follow a regular pattern in their distribution among natural numbers.
- Sum of Two Primes: The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers.
- Twin Primes: Twin primes are prime numbers that have a difference of 2, such as (3, 5), (11, 13), (17, 19), and so on.
Prime Numbers and Factors
Prime numbers are unique because they only have two factors: 1 and themselves. This property makes them fundamental in number theory and various mathematical applications. Some points about prime numbers and their factors:
- Prime Factorization: Every integer greater than 1 can be expressed uniquely as a product of prime numbers. This is known as prime factorization. For example, the prime factorization of 24 is 2 x 2 x 2 x 3.
- Factor Pairs: The factors of a number come in pairs. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. These factors can be paired up: 1 x 24, 2 x 12, 3 x 8, and 4 x 6.
- Prime Factorization Example: Let’s take the number 56. Its prime factorization is 2 x 2 x 2 x 7, where 2 and 7 are prime numbers. So, 56 has four factors: 1, 2, 7, and 56.
- Prime Factors and Composite Numbers: A composite number is a number that has more than two factors. Prime factors are the prime numbers that multiply together to give a composite number.
- Common Factors: When comparing two or more numbers, common factors are factors that are shared by all the numbers. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.
- Greatest Common Divisor (GCD): The greatest common divisor of two numbers is the largest positive integer that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.
- Least Common Multiple (LCM): The least common multiple of two numbers is the smallest positive integer that is a multiple of both numbers. For example, the LCM of 12 and 18 is 36.
How to Find Prime Numbers?
Here, we use three different approaches to find Prime Numbers with detailed explanation.
Approach 1: Sieve of Eratosthenes
To find prime numbers mathematically, you can use various methods. One of the most common approaches is the Sieve of Eratosthenes, which is an ancient algorithm for finding all prime numbers up to a specified integer. Here’s an explanation of how it works:
- Create a List: Start by creating a list of consecutive integers from 2 to the maximum number you want to check for primality.
- Start with the First Prime: The first prime number is 2. Mark 2 as a prime number and cross out all multiples of 2 from the list (4, 6, 8, 10, etc.).
- Move to the Next Unmarked Number: The next unmarked number in the list is 3. Mark 3 as a prime number and cross out all multiples of 3 from the list (6, 9, 12, 15, etc.).
- Repeat: Continue this process, marking the next unmarked number as a prime and crossing out its multiples, until you have processed all numbers in the list up to the square root of the maximum number.
- Identify Primes: The numbers that remain unmarked in the list after this process are prime numbers.
Approach 2: Using Formula 6n + 1 or 6n – 1
We can use 6n + 1 or 6n – 1 formula to find the prime numbers. This formula will not work for 2, and 3 because 2 and 3 smallest prime numbers and they are consecutives.
Note: [6n + 1] or [6n – 1] gives the prime number i.e. sometimes both [6n + 1] and [6n – 1] will be the prime number, and sometimes only one from 6n + 1 or 6n – 1.
Examples:
6 * 1 - 1 = 5 [For n = 1]
6 * 1 + 1 = 5 [For n = 1]
6 * 2 - 1 = 11 [For n = 2]
6 * 2 + 1 = 13 [For n = 2]
6 * 3 - 1 = 17 [For n = 3]
6 * 3 + 1 = 19 [For n = 3]
6 * 4 - 1 = 23 [For n = 4]
6 * 4 + 1 = 25 [For n = 4] - Multiple of 5 (Not Prime Number)
6 * 5 - 1 = 29 [For n = 5]
6 * 5 + 1 = 31 [For n = 5]
. . .
Approach 3: Using Formula n2 + n + 41, where n = 0, 1, 2, …, 39 (Works for Greater Then 40)
To get the prime number greater than 40, we can use the formula n2 + n + 41, where n = 0, 1, 2, …, 39.
Examples:
1^2 - 1 + 41 = 41 [For n = 1]
2^2 - 2 + 41 = 43 [For n = 2]
3^2 - 3 + 41 = 47 [For n = 3]
4^2 - 4 + 41 = 53 [For n = 4]
. . .
Is 1 a Prime Number?
No, 1 is not a prime number. By definition, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Since 1 has only one positive divisor (itself), it does not meet the criteria to be classified as a prime number.
Examples on Prime Numbers
Example 1: Is 21 a Prime Number?
Solutions: No, 21 is not a prime number because 21 is divided by 1, 3, 7, and 21.
Example 2: Is 29 a Prime Number?
Solutions: Yes, 29 is a prime number because 29 is divided by 1, and 29 i.e. 29 has only two factors 1 and 29 so it is prime number.
Alternative Solution –
29 = 6 * 5 – 1 [6n – 1, where n = 5], It means 29 is a prime number.
Example 3: Check 59 is a Prime Number or Not?
Solutions: Here, we use 6n – 1 formula to check 59 is prime number or not.
So, 59 = 6 * 10 – 1 [n = 10]
= 60 – 1
= 59
It follows the formula 6n – 1, so 59 is a prime number.
Example 4: How to First 10 Prime Numbers?
Solutions: The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
What are Co-Prime Numbers?
Co-prime numbers, also known as relatively prime or mutually prime numbers, are numbers that have no common factors other than 1. In other words, two numbers are co-prime if their greatest common divisor (GCD) is 1. For example, 8 and 15 are co-prime because the only positive integer that divides both 8 and 15 is 1. Co-prime numbers are important in number theory and have applications in various fields, including cryptography and computer science.
Frequently Asked Questions on Prime Numbers
Here are some frequently asked questions about prime numbers:
Q1. What is the definition of a prime number?
Ans. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Q2. What are the first few prime numbers?
Ans. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on.
Q3. How can you determine if a number is prime?
Ans. One way to determine if a number is prime is to check if it is divisible by any integer from 2 to the square root of the number. If it is not divisible by any of these integers, then it is prime.
Q4. Are there infinitely prime numbers?
Ans. Yes, there are infinitely many prime numbers. This was proven by the ancient Greek mathematician Euclid.
Q5. What is the twin prime conjecture?
Ans. The twin prime conjecture states that there are infinitely many pairs of prime numbers that are only two numbers apart, such as (3, 5), (11, 13), (17, 19), and so on.
Q6. What is the Sieve of Eratosthenes?
Ans. The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a specified integer.
Q7. What are composite numbers?
Ans. Composite numbers are natural numbers greater than 1 that are not prime, meaning they have divisors other than 1 and themselves.