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.
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For example, 7 is a divisor of 42 because 42 / 7 = 6. It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42. It usually written with a vertical bar between the two numbers, like 7  42. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
In general, it is said m  n (read as "m divides n") for nonzero 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 it is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 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). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.
A divisor of n that is not 1, −1, n, or − n (which are trivial divisors) is known as a nontrivial divisor; numbers with nontrivial divisors are known as composite numbers, while units (1 and 1) and prime numbers have no nontrivial divisors.
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.
There are properties which allow one to recognize certain divisors of a number from the number's digits.
For example, the set A = {1,2,3,4,5,6,10,12,15,20,30,60} of all positive divisors of 60, partially ordered by divisibility, has the Hasse diagram:
There are some elementary rules:
The vertical bar used is a Unicode "Divides" character, code point U+2223. Its negated symbol is ∤. In an ASCIIonly environment, the standard vertical bar "", which is slightly shorter, is often used.
The following property is important:
Also is useful fact that if p is a prime number and p  ab then p  a or p  b.
A positive divisor of n which is different from n is called a proper divisor or an aliquot part of n. A number that 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. 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. 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 the sum of their proper divisors are said to be abundant, while numbers greater than that sum are said to be deficient.
The total number of positive divisors of n is a multiplicative function d(n) (e.g. ). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. ). Both of these functions are examples of divisor functions.
If the prime factorization of n is given by
then the number of positive divisors of n is
and each of the divisors has the form
where for each
It can be shown that for any natural n the inequality holds.
Also it can be shown ^{[1]} that
One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about lnn.
The relation of divisibility turns the set N of nonnegative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest 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.
The generalization can be said to be the concept of divisibility in any integral domain.
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.
Finding one or more factors of a given number is called factorization.
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 mn (read: m divides n) for nonzero 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.)
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.
A divisor of n that is not 1, −1, n or −n is known as a nontrivial divisor; numbers with nontrivial divisors are known as composite numbers, while prime numbers have no nontrivial divisors.
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.
There are properties which allow one to recognize certain divisors of a number from the number's digits.
