Divisor Rules For Small Divisors Further Notions And Facts

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Number theory

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.

1 Rules for small divisors

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

A divisibility rule is a rule you can use to determine a number's divisibility by another number. In decimal, the divisibility rules are:

2 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).

3 Generalization

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

4 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 divisor s and Weil divisor s. 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.

5 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

6 External links

Factoring Calculator -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.

Arithmetic

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