Prímfaktorizációja

Prímfaktorizációja, often translated as prime factorization, is the process of expressing a composite number as a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, and 11 are prime numbers.

Every composite number can be uniquely represented as a product of prime numbers, regardless of the order of the factors. This fundamental concept is known as the Fundamental Theorem of Arithmetic. For instance, the prime factorization of 12 is 2 x 2 x 3, or 2^2 x 3. The prime factorization of 30 is 2 x 3 x 5.

The process of finding the prime factorization typically involves repeatedly dividing the number by the smallest possible prime divisor until the quotient is 1. For example, to find the prime factorization of 72:

72 ÷ 2 = 36

36 ÷ 2 = 18

18 ÷ 2 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

So, the prime factorization of 72 is 2 x 2 x 2 x 3 x 3, or 2^3 x 3^2.

Prime factorization has numerous applications in number theory and computer science, including finding the greatest common divisor (GCD) and the least common multiple (LCM) of two or more numbers, simplifying fractions, and in cryptography, particularly in algorithms like RSA. The difficulty of finding the prime factorization of very large numbers is the basis for the security of many cryptographic systems.

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