Divisor

For the second operand of a division, see division (mathematics). In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

For divisors in algebraic geometry, see divisor (algebraic geometry). In Algebraic geometry, divisors are a generalization of subvarieties of Algebraic varieties; two different generalizations are in common use Cartier divisors and

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left

1 Explanation
2 Further notions and facts
3 Divisibility of numbers
4 Generalization
5 References
6 See also
7 External links

Explanation

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 or 7 is a factor of 42 and we usually write 7 | 42. In Mathematics, a multiple of an Integer is the product of that integer with another integer For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

In general, we say m|n (read: m divides n) for non-zero integers m and n iff there exists an integer k such that n = km. ↔ Thus, divisors can be negative as well as positive, although often we restrict our attention to positive divisors. A negative number is a Number that is less than zero, such as −2 (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but one would usually mention only the positive ones, 1, 2, and 4. )

1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). In Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd. In Mathematics, the parity of an object states whether it is even or odd In Mathematics, the parity of an object states whether it is even or odd

A divisor of n that is not 1, −1, n or −n (which are trivial divisors) is known as a non-trivial divisor; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial divisors. A composite number is a positive Integer which has a positive Divisor other than one or itself In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

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. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the

There are properties which allow one to recognize certain divisors of a number from the number's digits. A divisibility rule is a method that can be used to determine whether a number is evenly divisible by other numbers

Further notions and facts

Some elementary rules:

If a | b and a | c, then a | (b + c), in fact, a | (mb + nc) for all integers m, n.
If a | b and b | c, then a | c. (transitive relation)
If a | b and b | a, then a = b or a = −b. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b

The following property is important:

If a | bc, and gcd(a,b) = 1, then a | c. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero (Euclid's lemma)

A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. Euclid's lemma ( Greek) is a generalization of Proposition 30 of Book VII of Euclid's Elements. In Mathematics, an aliquot part (or simply aliquot) of an integer is any of its Integer Proper divisors For instance 2 is an (A number which does not evenly divide n, but leaves a remainder, is called an aliquant part of n. In Mathematics, an aliquot part (or simply aliquot) of an integer is any of its Integer Proper divisors For instance 2 is an )

An integer n > 1 whose only proper divisor is 1 is called a prime number. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.

Any positive divisor of n is a product of prime divisors of n raised to some power. In Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder This is a consequence of the Fundamental theorem of arithmetic. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written

If a number equals the sum of its proper divisors, it is said to be a perfect number. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding Numbers less than the sum of their proper divisors are said to be abundant; while numbers greater than that sum are said to be deficient. In Mathematics, an abundant number or excessive number is a number n for which σ ( n) > 2 n. In Mathematics, a deficient number or defective number is a number n for which σ ( n) n.

The total number of positive divisors of n is a multiplicative function d(n) (e. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative 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)). Both of these functions are examples of divisor functions. In Mathematics, and specifically in Number theory, a divisor function is an Arithmetical function related to the Divisors of an Integer

If the prime factorization of n is given by

$n = p_1^{\nu_1} \, p_2^{\nu_2} \cdots p_k^{\nu_k}$

then the number of positive divisors of n is

$d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1),$

and each of the divisors has the form

$p_1^{\mu_1} \, p_2^{\mu_2} \cdots p_k^{\mu_k}$

where $0 \le \mu_i \le \nu_i$ for each $0 \le i \le k$ .

One can show ^[1] that

$d(1)+d(2)+ \cdots +d(n) = n \ln n + (2 \gamma -1) n + O(\sqrt{n}).$

One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about $ln n$ .

Divisibility of numbers

The relation of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. A negative number is a Number that is less than zero, such as −2 In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' 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. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero In Arithmetic and Number theory, the least common multiple or lowest common multiple ( lcm) or smallest common multiple of two This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z. In the mathematical area of Order theory, every Partially ordered set P gives rise to a dual (or opposite) partially ordered set which In Mathematics, the lattice of subgroups of a group G is the lattice whose elements are the Subgroups of G with the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

Generalization

One can talk about the concept of divisibility in any integral domain. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such Please see that article for the definitions in that setting.

References

^ Hardy, G. H.; E. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 M. Wright (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press, 264. ISBN 0-19-853171-0.

External links

Online Number Factorizer Instantly factors numbers up to 17 digits long
Factoring Calculator -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number. The tables contain the prime factorization of the Natural numbers from 1 to 1000 The tables below list all of the Divisors of the numbers 1 to 1000 In Number theory an arithmetic function or arithmetical function is a Function defined on the set of Natural numbers (i A divisibility rule is a method that can be used to determine whether a number is evenly divisible by other numbers In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, and specifically in Number theory, a divisor function is an Arithmetical function related to the Divisors of an Integer
downloadable factor program for factoring up to 18 digit numbers

Dictionary

-noun

(arithmetic) A number or expression that another is to be divided by. Eg. in "42 ÷ 3" the divisor is the 3.
An integer that exactly divides another integer an integer number of times.

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