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 42 is a multiple of 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.
The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient.
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 7 iff the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2×4 = 28 is divisible by 7)
- 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
Some elementary rules:
- If a|b and a|c, then a|(b+c).
- If a|b and b|c, then a|c.
- If a|b and b|a, then a=b or a=-b.
A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which does not evenly divide n', but leaves a remainder, is called an aliquant part of n.)
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.
If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than that sum are said to be deficient, while numbers greater than that sum are said to be abundant.
The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)).
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.
If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d. The rules for d=3 and d=9 given above are special cases of this result (b=10).
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
