Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 7 divides 42 and we usually write 7 | 42. Divisors can be positive or negative. The positive divisors of 42 are {1, 2, 3, 6, 7, 14, 21, 42}.

Some special cases: 1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd.

Table of contents

1 Rules for small divisors
2 Further notions and facts
3 Generalization
4 Divisors in Algebraic Geometry
5 See also

Rules for small divisors

There are some rules which allow to recognize small divisors of a number from the number's decimal digits:

a number is divisible by 2 iff the last digit is divisible by 2
a number is divisible by 3 iff the sum of its digits is divisible by 3
a number is divisible by 4 iff the number given by the last two digits is divisible by 4
a number is divisible by 5 iff the last digit is 0 or 5
a number is divisible by 6 iff it is divisible by 2 and by 3
a number is divisible by 8 iff the number given by the last three digits is divisible by 8
a number is divisible by 9 iff the sum of its digits is divisible by 9
a number is divisible by 10 iff the last digit is 0
a number is divisible by 11 iff the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)

Further notions and facts

A positive divisor of n which is different from n is called a proper divisor. An integer n > 1 whose only proper divisor is 1 is called a prime number.

Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the Fundamental theorem of arithmetic.

The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42)=8). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42)=96).

The relation | of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.

Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.

Divisors in Algebraic Geometry

In algebraic geometry, the word "divisor" is used to mean something rather different. Divisors are a generalization of subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors. The concepts agree on nonsingular varieties over algebraically closed fields. Any Weil divisor is a locally finite linear combination of irreducible subvarieties of codimension one. To every Cartier divisor D there is an associated line bundle denoted by [D], and the sum of divisors corresponds to tensor product of line bundles.