diviseurs

Diviseurs, in number theory, are integers that divide another integer exactly. For a positive integer n, a divisor d is an integer such that n is divisible by d, equivalently n mod d = 0. Divisors are typically considered among positive integers, though every positive divisor d has a corresponding negative divisor −d. The positive divisors of n form a finite, symmetric set: if d is a divisor of n, then n/d is also a divisor of n.

Prime and special divisors: among the divisors of n, the unit 1 is a divisor of every n, and n itself is a divisor of n. A prime divisor is a divisor that is prime. The divisors of n reflect its factorization and are used to assess properties such as primality, divisibility, and the structure of n’s multiples.

Factorization and divisor counts: by the fundamental theorem of arithmetic, if n = p1^a1 p2^a2 ... pk^ak is the prime factorization of n, then the number of positive divisors of n is d(n) = (a1+1)(a2+1)...(ak+1). The sum of the positive divisors is sigma(n) = ∏i (p_i^{a_i+1} − 1)/(p_i − 1).

Examples and applications: for n = 28 = 2^2 · 7, the positive divisors are 1, 2, 4, 7, 14, 28. Divisors are used in exploring perfect numbers (where the sum of proper divisors equals the number), amicable pairs, and in various algorithms for factorization and divisibility testing. In computational contexts, divisors can be found by factoring n and generating all products of the prime powers, or by trial division up to sqrt(n).