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Wikipedia / Divisor

In mathematics, a divisor of an integer also called a factor is a multiple is divisible or evenly divisible leaves no remainder.

The concept of a divisor is extended, with the same definition, to elements of any ring; see Divisibility (ring theory).

Definition

An integer This is written as

is permitted to be zero:

General

Divisors can be negative as well as positive, although often the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.

1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.

are known as the trivial divisors that is not a trivial divisor is known as a non-trivial divisor (or strict divisor). A nonzero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.

There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits.

Examples

  • 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.

  • The non-trivial divisors of 6 are 2, 2, 3, 3.

  • The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

  • The set partially ordered by divisibility, has the Hasse diagram:

Further notions and facts

There are some elementary rules:

This is called Euclid's lemma.

is called a or an but leaves a remainder is sometimes called an

whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.

is a product of prime divisors, each raised to some power. This is a consequence of the fundamental theorem of arithmetic.

is said to be perfect if it equals the sum of its proper divisors, deficient and abundant

is a multiplicative function are relatively prime). Both of these functions are examples of divisor functions.

If the prime factorization is given by

is

and each of the divisors has the form

Also,

is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n However, this is a result from the contributions of numbers with "abnormally many" divisors.

Division lattice

of non-negative integers into a partially ordered set that is 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 ∨ by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.

See also

Notes

Citations

References

  • ; section B

  • Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).