In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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 p^a divides n.

Two positive integers are coprime if and only if they have no prime factors in common. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than ↔ The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product. In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers It also follows from defining a and b as coprime if gcd(a,b)=1, so that gcd(1,b)=1 for any b>=1. Euclid's algorithm can be used to determine whether two integers are coprime without knowing their prime factors; the algorithm runs in a time that is polynomial in the number of digits involved. In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD

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. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written

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. In Number theory an arithmetic function or arithmetical function is a Function defined on the set of Natural numbers (i Different definitions exist depending on the specific field of application

Determining the prime factors of a number is an example of a problem frequently used to ensure cryptographic security in encryption systems; this problem is believed to require superpolynomial time in the number of digits- it is relatively easy to construct a problem that would take longer than the known age of the Universe to calculate on current computers using current algorithms. The age of the Universe is the time elapsed between the theory of the Big Bang and the present day

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 a unit)

See also

• Divisor
• Composite number
• Table of prime factors

External links

• A Javascript Prime Factor Calculator. Can handle numbers up to about 9×10¹⁵
• Java applet: Factorization using the Elliptic Curve Method finding factors with 20+ digits
• Lists of composites with prime factorization (first 100, first 1000, first 10,000, first 100,000, and first 1,000,000).