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Given two [[integer]]s ''d'' and ''a'', 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''.  Since 1 and -1 can divide any integer, they are said not to be ''proper'' divisors. The number 0 is not considered to be a divisor of ''any'' integer.
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Given two [[integer]]s ''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:
More examples:
:6 is a divisor of 24 since <math>6 \cdot 4 = 24</math>. (We stress that ''6 divides 24'' and ''6 is a divisor of 24'' mean the same thing.)
:6 is a divisor of 24 since <math>6 \cdot 4 = 24</math>. (We stress that "6 divides 24" and "6 is a divisor of 24" mean the same thing.)


:5 divides 0 because <math>5 \cdot 0 = 0</math>. In fact, every integer except zero divides zero.
:5 divides 0 because <math>5 \cdot 0 = 0</math>. In fact, every integer except zero divides zero.
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:7 divides 7 since <math>7 \cdot 1 = 7</math>.
:7 divides 7 since <math>7 \cdot 1 = 7</math>.


:1 divides 5 because <math> 1 \cdot 5 = 5</math>. It is, however, not a proper divisor.
:1 divides 5 because <math> 1 \cdot 5 = 5</math>.  


:-3 divides 9 because <math> (-3) \cdot (-3) = 9</math>
:&minus;3 divides 9 because <math> (-3) \cdot (-3) = 9</math>


:-4 divides -16 because <math>(-4) \cdot 4 = -16</math>
:&minus;4 divides &minus;16 because <math>(-4) \cdot 4 = -16</math>


:2 '''does not''' divide 9 because there is no integer k such that <math>2 \cdot k = 9</math>. Since 2 is not a divisor of 9, 9 is said to be an [[odd]] integer, or simply an [[odd]] number.
:2 '''does not''' divide 9 because there is no integer ''k'' such that <math>2 \cdot k = 9</math>. 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''.
*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 <math>0 \cdot k=0</math> 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.
*0 can never be a divisor of any number. It is true that 0&nbsp;&middot;&nbsp;d''k'' = 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.


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


====Notation====
==Related concepts==
If <math>d</math> is a divisor of a (we also say that <math>d</math> d ''divides'' <math>a</math>, this fact may be expressed by writing <math>d | a</math>. Similarly, if <math>d</math> does not divide <math>a</math>, we write <math>d \not| a</math>. For example, <math>4 | 12</math> but <math>8 \not| 12</math>.


===Related Concepts===
(If <math>d</math> is a divisor of <math>a</math> (<math>d | a</math>), we say <math>a</math> is a [[multiple]] of <math>d</math>. For example, since <math>4 | 12</math>, 12 is a multiple of 4. If both <math>d_1</math> and <math>d_2</math> are divisors of <math>a</math>, we say <math>a</math> is a common multiple of <math>d_1</math> and <math>d_2</math>. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique [[greatest common divisor]] of any two integers <math>a</math> and <math>b</math> written <math>\scriptstyle\operatorname{gcd}(a, b)</math> or, more commonly, <math>(a, b)</math>. 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 <math>a</math> and <math>b</math> is the smallest (positive) integer <math>m</math> such that <math>a | m</math> and <math>b | m</math>. Thus, the least common multiple of 12 and 9 is 36 (written <math>[12, 9] = 36</math>).


(If <math>d</math> is a divisor of <math>a</math> (<math>d | a</math>), we say <math>a</math> is a [[multiple]] of <math>d</math>. For example, since <math>4 | 12</math>, 12 is a multiple of 4. If both <math>d_1</math> and <math>d_2<math>are divisors of <math>a</math>, we say <math>a</math> is a common multiple of <math>d_1</math> and <math>d_2</math>. Ignoring the sign (i.e., only considering nonnegative integers), there is a unique [[greatest common divisor]] of any two integers <math>a</math> and <math>b</math> written <math>gcd(a, b)</math> or, more commonly, <math>(a, b)</math>. 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 <math>a</math> and <math>b</math> is the smallest (positive) integer <math>m</math> such that <math>a | m</math> and <math>b | m</math>. Thus, the least common multiple of 12 and 9 is 36 (written <math>[12, 9] = 36</math>).
==Abstract divisors==


===Further Reading===
In higher mathematics, the notion of divisor has been abstracted from the integers to the context of general [[commutative ring]]s.  In this setting, they might be termed [[divisor (ring theory)|abstract divisor]]s.
 
==Further reading==
*{{cite book
*{{cite book
|last = Scharlau
|last = Scharlau
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|publisher = Springer-Verlag
|publisher = Springer-Verlag
|date = 1985
|date = 1985
|isbn = 0-387-90942-7 }}
|isbn = 0-387-90942-7 }}[[Category:Suggestion Bot Tag]]
 
[[Category:CZ Live]]
[[Category:Mathematics Workgroup]]

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5 \cdot 0 = 0} . In fact, every integer except zero divides zero.
7 is a divisor of 49 since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7 \cdot 7 = 49} .
7 divides 7 since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7 \cdot 1 = 7} .
1 divides 5 because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \cdot 5 = 5} .
−3 divides 9 because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3) \cdot (-3) = 9}
−4 divides −16 because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-4) \cdot 4 = -16}
2 does not divide 9 because there is no integer k such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Notation

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is a divisor of a (we also say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , this fact may be expressed by writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d | a} . Similarly, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} does not divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \not| a} . For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 | 12} but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8 \not| 12} .

Related concepts

(If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is a divisor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d | a} ), we say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is a multiple of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . For example, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 | 12} , 12 is a multiple of 4. If both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2} are divisors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , we say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is a common multiple of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2} . Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\operatorname{gcd}(a, b)} or, more commonly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is the smallest (positive) integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a | m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b | m} . Thus, the least common multiple of 12 and 9 is 36 (written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [12, 9] = 36} ).

Abstract divisors

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

  • 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. 
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