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Encyclopedia > 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... The integers are commonly denoted by the above symbol. ... In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount left over— is also acknowledged. ...

Contents

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. 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. (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.) It has been suggested that this article or section be merged with Logical biconditional. ... A negative number is a number that is less than zero, such as −3. ...


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


A divisor of n that is not 1, −1, n or −n 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. Wikipedia does not yet have an article with this exact name. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ...


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, for tasks ranging from simple daily counting to advanced science and business calculations. ... 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 end result of a division problem. ...


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 divides 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.

The following property is important: In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

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.) In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... Euclids lemma is a generalisation of Proposition 30 of Book VII of Euclids Elements. ... In mathematics, an aliquot part (or simply aliquot) of an integer is any of its integer divisors. ... An aliquant part (or simply aliquant) is an integer that is not an exact divisor of a given quantity. ...


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


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. In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...


If a number equals the sum of its proper divisors, it is said to be a perfect number. 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, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ...


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)). In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...


If the prime factorization of n is given by In mathematics, the integer prime-factorization (also known as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. ...

n = p_1^{nu_1} , p_2^{nu_2} cdots p_n^{nu_n}

then the number of positive divisors of n is

d(n) = (nu_1 + 1) (nu_2 + 1) cdots (nu_n + 1),

and each of the divisors has the form

p_1^{mu_1} , p_2^{mu_2} cdots p_n^{mu_n}

where

forall i : 0 le mu_i le nu_i.

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. 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. A negative number is a number that is less than zero, such as −3. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... The lattice of subgroups of the dihedral group Dih4, represented as groups of rotations and reflections of a plane figure. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... The integers are commonly denoted by the above symbol. ...


Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting. In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...


See also

This table contains the integer factorization for the numbers from 1 to 1002. ... The tables below list all of the divisors of the numbers 1 to 1000. ... In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ... A divisibility rule is a method that can be used to determine whether a number divides other numbers. ...

External links

  • Online Number Factorizer Instanly 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.
  • webpage that has program for factoring up to 18 digit numbers

  Results from FactBites:
 
Greatest common divisor - Wikipedia, the free encyclopedia (764 words)
In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers.
The greatest common divisor is useful for reducing vulgar fractions to be in lowest terms.
If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b.
  More results at FactBites »

 
 

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