Prímtényezfelbontás

Prímtényezfelbontás, also known 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.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as a product of prime numbers, and that this representation is unique, apart from the order of the factors. This means that any composite number has a specific set of prime factors that, when multiplied together, result in the original number.

To find the prime factorization of a number, one can use various methods. A common approach is trial division, where you repeatedly divide the number by the smallest prime number that divides it evenly, until the number itself becomes a prime. For instance, to factor 12, you would divide by 2, getting 6. Then, divide 6 by 2, getting 3. Since 3 is prime, the prime factorization of 12 is 2 x 2 x 3, or 2^2 x 3.

Prime factorization has applications in various areas of mathematics, including finding the greatest common divisor (GCD) and least common multiple (LCM) of two or more numbers, simplifying fractions, and in number theory and cryptography. For example, to find the GCD of 12 and 18, you would find their prime factorizations: 12 = 2^2 x 3 and 18 = 2 x 3^2. The common prime factors are 2 and 3, each raised to the lowest power they appear in either factorization, so GCD(12, 18) = 2 x 3 = 6.