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Mathematics has a lot of wonderful concepts, one of which is the very basic concept of prime numbers. Whether you are starting a study into number theory or dusting off some math skills, one should know what a prime number really is. In this section, prime numbers definition and examples will be presented, answered two questions: what are prime numbers? And a list of prime numbers will be given for quick reference.
To begin with, let us understand what are prime numbers in maths. A natural number greater than 1 is called a prime number when it has exactly two factors: 1 and itself. In simpler words, it is not divisible evenly by any other number.
For instance:
2 can be considered a prime number since the only numbers which divide it evenly are 1 and 2.
Similarly, 3 is a prime number because it is divisible only by 1 and itself.
4, on the other hand, is not prime because it has three divisors: 1, 2, and 4.
Thus, prime numbers are basically "building blocks" of any kind of mathematics-whether multiplication or factorization.
According to the conventional definition of prime numbers:
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Those numbers which have more than two factors are composite numbers. For example, 6 is composite as it can be divided by 1, 2, 3, and 6.
If you are learning or teaching, a prime numbers list will be handy. Here is a list of prime numbers up 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
Notice that 2 is the only even prime number. Any other even number can be divided by 2, hence making it composite.
Examples of Prime Numbers
Some examples of prime numbers are:
17 → Factors: 1, 17 → Prime
29 → Factors: 1, 29 → Prime
35 → Factors: 1, 5, 7, 35 → Not Prime (Composite)
41 → Factors: 1, 41 → Prime
These examples reveal how prime numbers are extraordinary in that they cannot be expressed as the product of smaller whole numbers except by 1 and themselves.
Now that we know Prime Numbers in mathematics, let us study the properties:
Exactly Two Factors- Every prime number will have exactly two factors.
Special role in multiplication- Or any natural number greater than 1 is either a prime itself or can be expressed as a product of primes (which is known as prime factorization).
All other primes are odd- 2 is the smallest and only even prime number.
Infinite in number- There is no largest prime number, and they go on infinitely.
Prime numbers not only find use in school mathematics but also play a critical role in advanced applications:
1. Cryptography (online security) --> Modern encryption algorithms, such as RSA, depend on large prime numbers to ensure secure digital communication.
2. Numbering theory and mathematics --> Prime numbers are the basic elements for many theorems and mathematical proofs.
3. Computer science --> Prime numbers are used by certain algorithms for hashing, coding, and random number generation.
It is evident that prime numbers are of interest not only in theoretical terms but also in practical terms.
Some cues you can use to identify primes in a snap:
Big or small, always check divisibility by 2, 3, 5, or 7.
If divisible by any of these numbers-other than the number itself-it is not prime.
Keep the square root rule in mind: For a big number, you only need to check whether it is divisible by any number less than or equal to its square root.
Example: To check whether 59 is prime, test for divisibility by numbers up to √59(≈7.6). Since it is not divisible by 2, 3, 5, or 7, it is prime.
Quick rejection: Any number ending on 0, 2, 4, 5, 6, 8 is composite other than 2 and 5.
If you are intrigued beyond the simple list of primes, here are the probable primes after 100:
101, 103, 107, 109, 113, 127, 131, 137, 139… ad infinitum.
Numbers that are greater than one and can be factored by only two numbers: one and itself are called Prime Numbers
Definitions and examples of prime numbers help to understand that primes cannot be divided by any other number.
A list of prime numbers can ease the recognition and memorization process.
They are not just numbers for school exercises but prevail also over technology, cryptography, and ecological problems.
Prime numbers essentially become the mathematics spine. The more you go through it, the more fascinating patterns and applications you will find.
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