Prime factor
From Wikipedia, the free encyclopedia
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization.
For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n.
Two positive integers are coprime if and only if they have no prime factors in common. The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product.
The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicity. The fundamental theorem of arithmetic says that every positive integer has a unique prime factorization.
For a positive integer n, the number of prime factors of n and the sum of the prime factors of n (not counting multiplicity) are examples of arithmetic functions of n that are additive but not completely additive.
[edit] Examples
- The prime factors of 6 are 2 and 3 (6 = 2 × 3). Both have multiplicity 1.
- 5 has only one prime factor: itself (5 is prime). It has multiplicity 1.
- 100 has two prime factors: 2 and 5 (100 = 22 × 52). Both have multiplicity 2.
- 2, 4, 8, 16, etc. each have only one prime factor: 2. (2 is prime, 4 = 22, 8 = 23, etc.)
- 1 has no prime factors. (1 is the empty product)
