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*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}.

**divides**42Some 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 |

2 Further notions and facts 3 Generalization 4 Divisors in Algebraic Geometry 5 See also |

### Rules for small divisors

- 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.

### See also

- Table of prime factors -- A table of prime factors for 1-1000
- Table of divisors -- A table of prime and non-prime divisors for 1-1000