Divisor

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For the second operand of a division, see Division (mathematics).
For divisors in algebraic geometry, see Divisor (algebraic geometry).
"Aliquot part" redirects here. For other uses, see aliquot.

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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[edit] Explanation

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 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 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 non-trivial divisor; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial 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:

[edit] Further notions and facts

Sister project The Wikibook The Book of Mathematical Proofs has a page on the topic of

There are some elementary rules:

  • If a | b and a | c, then a | (b + c). In fact, a | (mb + nc) for all integers m and n.
  • If a | b and b | c, then a | c. This is the transitive relation.
  • If a | b and b | a, then a = b or a = − b.

The vertical bar used is a Unicode "Divides" character, code point U+2223. Its negated symbol is ∤. In an ASCII-only 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 f 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. d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. \sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7)). Both of these functions are examples of divisor functions.

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.

It can be shown that for any natural n inequality d(n) < 2 \sqrt{n} holds.

Also it can be shown [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 lnn.

[edit] 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 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.

[edit] Generalization

The generalization can be said to be the concept of divisibility in any integral domain.

[edit] Notes

  1. ^ Hardy, G. H.; E. M. Wright (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. pp. 264. ISBN 0-19-853171-0. 

[edit] References

[edit] External links

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