Definition of a Prime Number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it cannot be formed by multiplying two smaller natural numbers. For instance, 2, 3, 5, and 7 are prime numbers because they are only divisible by 1 and their own value.
Key Properties of Prime Numbers
Prime numbers possess unique properties that distinguish them from composite numbers. They are indivisible except by 1 and themselves, and every natural number greater than 1 is either prime or can be factored into primes (the Fundamental Theorem of Arithmetic). Additionally, 2 is the only even prime number, while all others are odd.
Examples of Prime Numbers
To identify a prime number, check for divisors up to its square root. For example, 13 is prime because it has no divisors other than 1 and 13: it is not divisible by 2, 3, or any number up to √13 (approximately 3.6). In contrast, 9 is not prime as it is divisible by 3 (3 × 3 = 9). Common primes include 2, 3, 5, 7, 11, 13, 17, and 19.
Importance and Applications of Prime Numbers
Prime numbers are essential in mathematics and real-world applications, serving as the foundation for factorization and encryption algorithms like RSA, which secure online communications. They appear in fields such as cryptography, computer science, and physics, enabling secure data transmission and solving complex computational problems.