In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder.
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 the maximum power of each prime factor that divides the integer exactly. 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.
Examples
- The prime factors of 6 are 3 and 2. (6 = 3 × 2)
- 5 has only one prime factor: itself. (5 is prime)
- 100 has two prime factors: 2 and 5. (100 = 22 × 52)
- 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)
See also