Divisor

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This editable Main Article is under development and subject to a disclaimer.

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This article is about mathematical divisors. For other uses of the term Divisor, please see Divisor (disambiguation).

Given two integers d and a, where d is nonzero, d is said to divide a, or d is said to be a divisor of a, if and only if there is an integer k such that dk = a. For example, 3 divides 6 because 3 * 2 = 6. Here 3 and 6 play the roles of d and a, while 2 plays the role of k. Though any number divides itself (as does its negative), it is said not to be a proper divisor. The number 0 is not considered to be a divisor of any integer.

More examples:

6 is a divisor of 24 since 6⋅4=24. (We stress that "6 divides 24" and "6 is a divisor of 24" mean the same thing.)

5 divides 0 because 5⋅0=0. In fact, every integer except zero divides zero.

7 is a divisor of 49 since 7⋅7=49.

7 divides 7 since 7⋅1=7.

1 divides 5 because 1⋅5=5.

−3 divides 9 because (−3)⋅(−3)=9

−4 divides −16 because (−4)⋅4=−16

2 does not divide 9 because there is no integer k such that 2⋅k=9. Since 2 is not a divisor of 9, 9 is said to be an odd integer, or simply an odd number.

* When d is non zero, the number k such that dk = a is unique and is called the exact quotient of a by d, denoted a/d.
* 0 can never be a divisor of any number. It is true that 0 * dk = 0 for any k, however, the quotient 0/0 is not defined, as any k would work. This is the reason 0 is excluded from being considered a divisor.

## Contents

* 1 Notation
* 2 Related concepts
* 3 Abstract divisors
* 4 Further reading

## Notation[edit]

If d is a divisor of a (we also say that d divides a, this fact may be expressed by writing d|a. Similarly, if d does not divide a, we write d|a. For example, 4|12 but 8|12.

## Related concepts[edit]

(If d is a divisor of a (d|a), we say a is a multiple of d. For example, since 4|12, 12 is a multiple of 4. If both d1 and d2 are divisors of a, we say a is a common multiple of d1 and d2. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers a and b written gcd⁡(a,b) or, more commonly, (a,b). The greatest common divisor of 12 and 8 is 4, the greatest common divisor of 15 and 16 is 1. Two numbers with a greatest common divisor of 1 are said to be relatively prime. Complementary to the notion of greatest common divisor is least common multiple. The least common multiple of a and b is the smallest (positive) integer m such that a|m and b|m. Thus, the least common multiple of 12 and 9 is 36 (written [12,9]=36).

## Abstract divisors[edit]

In higher mathematics, the notion of divisor has been abstracted from the integers to the context of general commutative rings. In this setting, they might be termed abstract divisors.

## Further reading[edit]

* Scharlau, Winfried; Opolka, Hans (1985). From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development. Springer-Verlag. ISBN 0-387-90942-7.