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# Divisor

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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.

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.

### 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 two sums of alternate digits differ by 0 or 11 (e.g. 1837 is divisible by 11 as 1+3 = 4; 8+7 = 15; 4 and 15 differ by 11.)

### 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.

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