Divisori

Divisori, or divisors, are integers that divide another integer exactly. In mathematics, a divisor d of a positive integer n satisfies n mod d = 0. The study typically focuses on positive divisors, and every positive integer n > 0 has a finite set of positive divisors, which always includes 1 and n; negative divisors are also considered in some contexts, yielding pairs of signs.

Prime divisors are divisors that are themselves prime; the prime divisors of n correspond to the distinct primes in its prime factorization. For example, the positive divisors of 12 are 1, 2, 3, 4, 6, and 12; its prime divisors are 2 and 3.

The Fundamental Theorem of Arithmetic implies that, if n = p1^a1 p2^a2 ... pk^ak is the prime factorization of n, then n has (a1+1)(a2+1)...(ak+1) positive divisors. The sum of all positive divisors is given by σ(n) = ∏ (p_i^{a_i+1}-1)/(p_i-1). Divisors can be enumerated from the factorization, and the set of divisors forms a lattice under divisibility. The divisor function d(n) counts the positive divisors, and the sum-of-divisors function σ(n) aggregates them; both are multiplicative: if gcd(m,n)=1 then d(mn)=d(m)d(n) and σ(mn)=σ(m)σ(n).

Divisors have several applications in number theory, including gcd and lcm computations, modular arithmetic, and the study of special numbers such as perfect numbers, where the sum of proper divisors equals the number itself, and abundant or deficient numbers, depending on whether that sum is greater or less than the number.

Historically, the concept is ancient and remains central in elementary and higher number theory, with related notions such as proper divisors, prime divisors, and divisor functions continuing to be studied.