Divider

In Mathematical, a dividing of a whole N , also called a factor of N if it is an entirety which also divides N without leaving a remains. For example, 7 is a divider of 42 because 42/7 = 6. We say also 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we generally write 7 | 42. The dividers can be positive or negative. The positive dividers of 42 are {1, 2,3,6,7,14,21,42}.

Certain special cases: 1 and -1 is the dividers of each entirety, and each entirety is a divider of 0. The divisible numbers by 2 are called even and those which are not it are called odd.

The name comes from the arithmetic operation of division: if a/b=c then has is the dividend , B the dividing , and C the Quotient .

Criteria for the small dividers

There exist certain rules which make it possible to recognize the small dividers of a number starting from the number of its decimal digits.

A Critère of divisibility is a criterion which you can use to determine the divisibility of a number by another number. Those are based on the congruence S modulo the sought divider.

For example, in decimal system, we want to know the criterion for divider 3.

10 is adequate to 1 modulo 3,100 is adequate to 1 modulo 3,…, 10^n is adequate to 1 modulo 3

Any number in being written Decimal system:

$an\; \backslash \; times\; 10^n\; +\; \backslash \; ldots\; +\; a2\; \backslash \; times\; 10^2\; +\; a1\; \backslash \; times\; 10^1\; +\; a0\; \backslash \; times\; 10^0\; \backslash ,$, it comes

$an\; \backslash \; times\; 1\; +\dots \; +\; a2\; \backslash \; times\; 1\; +\; a1\; \backslash \; times\; 1\; +\; a0\; \backslash \; times\; 1\; \backslash ,$

i.e. the sum of the Coefficient S of the powers of 10, i.e. the sum of the figures which compose the number, those must be divisible by 3 so that the number is divisible by 3.

Additional concepts and results

Some elementary rules:

• If has | B and has | C , then has |( bx + cy ).
• If has | B and B | C , then has | C .
• If has | B and B | has , then has = B or has = - B .

A positive divider of N which is different from N is called a strict dividing (or an aliquot left ) of N . (A number which also does not divide , but which leaves a remainder, is called an aliquant left N ).

A positive divider of N noncommonplace, i.e. different of N and 1 is called a clean dividing .

A Integer N > 1 which does not have a clean divider is called a Prime number.

Any divider of N is the product of prime factor of N with a certain power while exposing. This is a consequence of the fundamental Théorème of arithmetic the.

If a number is equal to the sum of its strict dividers, it is called a Perfect number. The numbers lower than this sum are known as defective , whereas the numbers larger than this sum are known as abundant .

The full number of positive dividers of N is a multiplicative Fonction D ( N ) (for example:

$d\; (42)\; =\; 8\; =\; 2\; \backslash \; times\; 2\; \backslash \; times\; 2\; =\; D\; (2)\; \backslash \; times\; D\; (3)\; \backslash \; times\; D\; (7)\; \backslash ,$).

The sum of the positive dividers of N is another multiplicative function $\backslash \; sigma\; (N)\; \backslash ,$ (for example:

$\backslash \; sigma\; (42)\; =\; 96\; =\; 3\; \backslash \; times\; 4\; \backslash \; times\; 8\; =\; \backslash \; sigma\; (2)\; \backslash \; times\; \backslash \; sigma\; (3)\; \backslash \; times\; \backslash \; sigma\; (7)\; \backslash ,$).

The relation | with divisibility the unit $\backslash \; mathbb\; \{NR\}$ with the Integers positive of a relation of a partial nature provides, which in fact a Ensemble partially ordered, makes of it in a complete distributive network. The largest element of this lattice is 0 and smallest is 1. The binary relation of upper limit $\backslash \; vee$ is given by PGCD and the binary relation of lower limit $\backslash \; wedge$ by PPCM. This network is isomorphous with dual network of the Sous-groupe S of the cyclic Groupe infinite $\backslash \; mathbb\; \{Z\}\; \backslash ,$.

If an integer N is written in a bases '' 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 figures is divisible by D . The criterion given for D =3 given higher is a particular case of this result. ( B =10).

See also: Arithmetic modular

If the Décomposition in factors first of N is given by

$N\; =\; p\_1^\; \{\backslash \; nu\_1\}\; \backslash ,\; p\_2^\; \{\backslash \; nu\_2\}\; \backslash \; cdots\; p\_n^\; \{\backslash \; nu\_n\}$

then the number of positive dividers of N is

$D\; (N)\; =\; (\backslash \; nu\_1\; +\; 1)\; (\backslash \; nu\_2\; +\; 1)\; \backslash \; cdots\; (\backslash \; nu\_n\; +\; 1),$

and each divider is form

$p\_1^\; \{\backslash \; mu\_1\}\; \backslash ,\; p\_2^\; \{\backslash \; mu\_2\}\; \backslash \; cdots\; p\_n^\; \{\backslash \; mu\_n\}$

where

$\backslash \; forall\; I:\; 0\; \backslash \; the\; \backslash \; mu\_i\; \backslash \; the\; \backslash \; nu\_i.$

Dividers in algebraic geometry

In algebraic Geometry, the “dividing” word is used in a rather different direction. The dividers are a generalization of the Sous-variété S of algebraic varieties; two different generalizations are of a common use, the dividing of Cartier and the dividing of Weil. The concepts are appropriate for the not-singular varieties algebraically of the closed fields. Any divider of Weil is a linear Combinaison locally finished irreducible subvarieties of codimension one. For each divider of Cartier D , there exists a associated Paquet of line indicated by ', and dividers corresponds to the tensorial Produit of the packages of lines summons it.

See too

• Table of the factors first -- A table of the factors first from 1 to 1000

• Table of the dividers -- A table of prime factors and not-first from 1 to 1000

External bonds

• Calculator Factoring - a calculator of factorization which posts the factors first and not first of a given number.

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