Prímfaktorizálás

Prímfaktorizálás, also known as prime factorization, is the process of breaking down a composite number into its constituent prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 1 that is not prime. Every composite number can be uniquely expressed as a product of prime numbers, a fundamental concept known as the Fundamental Theorem of Arithmetic.

To perform prímfaktorizálás, one can use a variety of methods. A common approach is trial division, where you repeatedly divide the number by the smallest prime number possible until you can no longer divide by that prime. Then, you move to the next smallest prime and repeat the process. This continues until the number is reduced to 1. For example, to factorize 12, you would divide by 2, getting 6. Then divide 6 by 2, getting 3. Finally, divide 3 by 3, getting 1. Thus, the prime factorization of 12 is 2 x 2 x 3, or 2^2 x 3.

Prímfaktorizálás has numerous applications in mathematics, including finding the greatest common divisor (GCD) and the least common multiple (LCM) of two or more numbers. It is also a crucial component in public-key cryptography, where the difficulty of factoring very large numbers forms the basis of secure communication systems. While simple for small numbers, prímfaktorizálás becomes computationally challenging for extremely large numbers, making it a cornerstone of modern cybersecurity.