Prime Numbers – Definition with Examples

Table of Contents

    Discovered by Eratosthenes, prime numbers are naturally occurring numbers that can be divided only by 1 and itself. Basically, prime numbers in maths are numbers with positive integers greater than one with only two factors: one and itself. 

    This article gives in-depth details on what is a prime number, properties of a prime number, prime numbers chart, easy ways to find prime numbers, prime numbers list, prime numbers up to 100, and more topics.

    What Are Prime Numbers?

    A prime number is any number that is divisible by only 1 and itself. This means a number becomes a prime number if it can be divided by 1 and itself without leaving any remainder. Prime numbers are generally greater than 1 and have just two factors.

    For example, 3 is a prime number because it can only be divided by 1 and itself, 3 without leaving a remainder. The number 3 has only two factors, hence it fulfills all the requirements of a prime number. On the contrary, 8 is divisible by 1, 2, 4 and 8. Hence, 8 cannot be classified as a prime number since it has 4 factors.

    Properties of Prime Numbers

    Prime numbers have a number of recognizable properties or features. They are:

    • With the exception of 2, all prime numbers are odd numbers. This feature makes 2 the only even prime number.
    • A prime number is any number greater than 1, which is divisible by another prime number 1 and itself.
    • All prime numbers have exactly two factors: 1 and itself.
    • Two prime numbers are always co-prime of one another.
    • Every composite number can be uniquely factored to prime factors.

    Prime Numbers Chart

    Prime numbers chart, also known as Eratosthenes chart, is a numerical table that records all prime numbers and prime factors. 

    Prime numbers charts are one of the quickest ways of identifying prime numbers. It comes with prime numbers already identified, as well as arranged in rows and columns with the prime numbers highlighted.

    Easy Ways to Find Prime Numbers

    There are a number of ways to find a prime number:

    Sieve of Eratosthenes

    Named after a Greek mathematician, this method simplifies the identification of prime numbers and the creation of a prime numbers list. To use this method to identify prime numbers up to 100, create a chart from 2 to 100, circle 2, mark off its multiples, circle 3, mark off its multiples, and continue the process with uncircled numbers and their multiples until all circled numbers are prime and all marked-off numbers are composite.

    Prime Factorization

    Factorization is a primary method for identifying prime numbers. Factors of a number are the numbers that divide it without a remainder. If a number has more than two factors, it is not prime. For instance, 8 has four factors, so it is not a prime number. However, 7 is only divisible by 1 and itself, which makes it prime.

    Prime Formulas

    Prime formulas help identify large prime numbers effectively. The first formula checks the unit place and eliminates numbers ending in 0, 2, 4, 6, 8, and 5. The second formula adds up the digits and checks if the sum is divisible by 3. Finally, the third formula finds the square root and divides it by all prime numbers less than its value. If divisible, it’s not prime, if not, it’s prime.

    List of Prime Numbers

    Let’s take a look at the list of some prime numbers:

    List of Prime Numbers between 1 and 100

    Between 1 and 100, there are only 25 prime numbers. These include 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.

    List of Prime Numbers between 1 and 200

    Between 1 and 200, there are 46 prime numbers. There include 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 between 1 and 1000

    There are 168 prime numbers between 1 and 1000. There 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, 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 and Composite Numbers

    There are a number of features that differentiate a prime number from a composite number. A prime number definition states it is a natural number greater than 1 with just two factors. Composite numbers on the other hand, are numbers greater than 1 with more than two factors, that is, they are divisible by more than two factors.

    For example, 4 is a composite number because it is divisible by more than two factors: 1, 2, and 4. The least prime number is 2, while the least composite number is 4. The examples of composite numbers include 4, 6, 8, 9, 10, 12, and more.

    Prime Numbers and Co-Prime Numbers

    While prime numbers are considered singularly, co-prime numbers are examined in pairs. A pair of numbers with no common factor except for 1 are considered co-prime numbers. 

    Co-prime numbers can either be prime or composite subject to the condition that the greatest common factor (GCF) of the numbers is 1. 

    The examples of a co-prime number are 5 and 9. Factors of 5 include 1 and 5. Factors of 9 include 1, 3, and 9. The common factor between 5 and 9 is 1; hence, both qualify to be categorized as co-prime numbers. 

    Solved Examples on Prime Numbers

    To further understand what a prime number is, below are some prime number examples:

    Example 1:

    What is the smallest prime number greater than 20?

    Solution: 

    The smallest prime number greater than 20 is 23. To see this, we can simply check all the numbers after 20 that are greater than 1 and not divisible by any other number except 1 and itself. 

    Starting from 21, we can see that it is divisible by 3, 22 is divisible by 2 and 11, and finally, we arrive at 23 which is a prime number.

    Answer: 23. 

    Example 2:

    Find all prime numbers between 30 and 50.

    Solution:

    To find all prime numbers between 30 and 50, we can check each number from 31 to 49 to see if it is divisible by any prime number less than or equal to its square root.

    We can use trial division to check each number for divisibility by the primes 2, 3, 5, and 7. 

    We can start with 31 and check if it is divisible by any of these primes. We can see that it is not divisible by any of them, so 31 is a prime number. 

    Similarly, we can check 37, 41, 43, and 47 and see that they are also prime. 

    Answer: Therefore, the prime numbers between 30 and 50 are 31, 37, 41, 43, and 47.

    Example 3:

    What is the largest prime factor of 720?

    Solution: 

    To find the largest prime factor of 720, we can use the prime factorization of 720 which is 2^4 x 3^2 x 5. 

    We can see that the largest prime factor of 720 is 5 since it is the largest prime number among its prime factors. 

    Answer: Therefore, the answer is 5.

    Frequently Asked Questions

    Is 1 a prime number?

    1 is not a prime number. This is because 1 is only divisible by itself which means 1 has just a single factor. To be qualified as a prime number, a number must have two factors.

    Can a prime number be negative?

    No, a prime number cannot be negative. By definition, all prime numbers have no positive divisors apart from 1 and the number itself.

    What is the difference between a prime number and a co-prime number?

    Co-prime numbers are examined in pairs. Both numbers must have 1 as a common factor. A prime number definition does not subject prime numbers to this condition and are considered as a single number. 

    What is the smallest prime number?

    The smallest prime number is 2.

    What is the largest known prime number?

    As of February 2023, the largest prime number is 282, 589, 9339 – 1.

    Which numbers are not prime numbers?

    0 and 1.

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