Divisor (number theory)

Definition and basic notation

When we speak of divisors we often distinguish between positive divisors and negative divisors: if m divides n then so does −m, because (−m)(−k)=n. The positive divisors of n are commonly listed for classification and counting. A proper divisor of n is any positive divisor less than n; it excludes n itself. A prime number is an integer greater than 1 whose only positive divisors are 1 and itself, while a composite number has additional divisors.

Common properties and examples

• If a divides b and b divides c then a divides c (transitivity).
• If a divides b then a divides any multiple of b.
• The greatest common divisor (gcd) of two integers is the largest integer that divides both; it can be found by repeated division (the Euclidean algorithm).
• Divisibility tests give quick checks for small primes—for example, divisibility rules for 2, 3, 5, 9 and 11.

Related concepts

Factorization is the process of expressing a number as a product of divisors, typically primes. The study of divisors connects to the divisor function (which counts divisors), the structure of the ring of integers, and to greatest common divisors and least common multiples. For background on integers in mathematics see mathematics and for a notation reference see integer n and discussion of remainders at remainder.

Uses, history and significance

Divisors are fundamental in solving Diophantine equations, analyzing arithmetic functions, and in applications such as cryptography where factoring large integers is computationally difficult. Historically, ideas about divisibility and prime numbers date back to ancient Greek mathematics and were formalized by Euclid; modern algorithmic developments enable fast computations and have practical importance in computer science. For topics on factorization algorithms and their applications see factorization and for the definition of primes see prime number.

Understanding divisors gives a simple, unifying language for many results in number theory and elementary arithmetic. Whether counting divisors, testing primality, or computing gcds, the basic idea—that one integer fits evenly into another—remains a foundational tool.