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.
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2 Further notions and facts 3 Generalization 4 Divisors in algebraic geometry 5 See also 6 External links |
There are some rules which allow to recognize small divisors of a number from the number's decimal digits:
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.
A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A part of n which does not evenly divide it, 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 divisorss 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 the sum of their divisors are 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.
One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.
Let C, D and B be natural numbers, being B>D, Let C digits be written in a B base. For any D that complies with (B mod D = 1), if the D modulus of the sumatorium of C digits is 0, this implies that C mod D = 0.
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.
Rules for small divisors
Example: Is 161 divisible by 7?
Last digit= 1 and 1X2=2
Subtract 2 from remainder of number, 16
16-2 = 14. 14 is divisible by 7, therefore 161 is also.
Further notions and facts
Generalization
Divisors in algebraic geometry
See also
External links