Teilers

Teiler is the German term for divisor in number theory. A number a is a Teiler (divisor) of n if n is a multiple of a, i.e., n mod a = 0 or a divides n. In the context of integers, divisors can be positive or negative; most elementary discussions focus on positive divisors.

Positive divisors of n are the numbers d > 0 such that there exists k with n = d·k. The set of positive divisors is finite. The number of positive divisors is denoted d(n) or τ(n). If the prime factorization of n is n = p1^a1 p2^a2 ... pk^ak, then d(n) = (a1+1)(a2+1)...(ak+1). The sum of positive divisors is σ(n) = ∏ (p_i^{a_i+1}-1)/(p_i-1).

Example: n = 12 has positive divisors 1, 2, 3, 4, 6, 12. Proper divisors are those less than n: 1, 2, 3, 4, 6. The prime divisors of 12 are 2 and 3.

Every integer greater than 1 has a unique prime factorization, by the Fundamental Theorem of Arithmetic. The divisors of n correspond to selecting exponents e_i with 0 ≤ e_i ≤ a_i in the prime factorization, yielding ∏ (a_i+1) divisors.

Algorithms for listing divisors often test integers up to the square root of n; each divisor d pairs with n/d, which is also a divisor. Negative divisors are simply the negatives of the positive divisors.

Applications of the concept of divisors include gcd and lcm calculations, the study of arithmetic functions such as φ(n) (Euler’s totient function), and various factorization and number-theory problems.