Divisor

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

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

[edit intro]

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 {\displaystyle 6\cdot 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 {\displaystyle 5\cdot 0=0} . In fact, every integer except zero divides zero.

7 is a divisor of 49 since  7 ⋅ 7 = 49 {\displaystyle 7\cdot 7=49} .

7 divides 7 since  7 ⋅ 1 = 7 {\displaystyle 7\cdot 1=7} .

1 divides 5 because  1 ⋅ 5 = 5 {\displaystyle 1\cdot 5=5} .

−3 divides 9 because  ( − 3 ) ⋅ ( − 3 ) = 9 {\displaystyle (-3)\cdot (-3)=9}

−4 divides −16 because  ( − 4 ) ⋅ 4 = − 16 {\displaystyle (-4)\cdot 4=-16}

2 does not divide 9 because there is no integer k such that  2 ⋅ k = 9 {\displaystyle 2\cdot 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 {\displaystyle d} is a divisor of a (we also say that  d {\displaystyle d} divides a {\displaystyle a} , this fact may be expressed by writing  d | a {\displaystyle d|a} . Similarly, if  d {\displaystyle d} does not divide  a {\displaystyle a} , we write  d ⧸ | a {\displaystyle d\not |a} . For example,  4 | 12 {\displaystyle 4|12} but  8 ⧸ | 12 {\displaystyle 8\not |12} .

## Related concepts[edit]

(If  d {\displaystyle d} is a divisor of  a {\displaystyle a} ( d | a {\displaystyle d|a} ), we say  a {\displaystyle a} is a multiple of  d {\displaystyle d} . For example, since  4 | 12 {\displaystyle 4|12} , 12 is a multiple of 4. If both  d 1 {\displaystyle d_{1}} and  d 2 {\displaystyle d_{2}} are divisors of  a {\displaystyle a} , we say  a {\displaystyle a} is a common multiple of  d 1 {\displaystyle d_{1}} and  d 2 {\displaystyle d_{2}} . Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers  a {\displaystyle a} and  b {\displaystyle b} written  gcd ⁡ ( a , b ) {\displaystyle \scriptstyle \operatorname {gcd} (a,b)} or, more commonly,  ( a , b ) {\displaystyle (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 {\displaystyle a} and  b {\displaystyle b} is the smallest (positive) integer  m {\displaystyle m} such that  a | m {\displaystyle a|m} and  b | m {\displaystyle b|m} . Thus, the least common multiple of 12 and 9 is 36 (written  [ 12 , 9 ] = 36 {\displaystyle [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.