Divisors

Divisors of an integer n are the integers d such that d divides n, meaning there exists an integer m with n = d m. In most contexts, divisors are taken to be positive, and the positive divisors of n form a finite set when n ≠ 0. If n ≠ 0, d is a divisor of n exactly when n/d is an integer. If n = 0, every nonzero integer d satisfies 0 = d·0, so all nonzero integers are divisors of 0.

The number of positive divisors is denoted d(n) or tau(n). If the prime factorization n = p1^a1 p2^a2 ... pk^ak, then d(n) = (a1+1)(a2+1)…(ak+1). Divisors include 1 and n, and, when negative divisors are allowed, also -1 and -n. For example, the positive divisors of 12 are 1, 2, 3, 4, 6, 12.

Divisibility is a partial order: if d|n and e|d, then e|n. The sum-of-divisors function sigma(n) adds all positive divisors of n; a number is called perfect if sigma(n) = 2n. Proper divisors are divisors other than the number itself, with 1 usually included for n > 1.

Divisors have applications in factorization, arithmetic functions, and number theory. The concept extends to polynomials and other rings: a divides b if there exists c with b = a c.