Divisor: Difference between revisions

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*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 <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.


====Notation====
====Notation====
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>.
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>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>).


===Further Reading===
===Further Reading===

Revision as of 19:28, 31 March 2007

Given two integers 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.

More examples:

6 is a divisor of 24 since . (We stress that 6 divides 24 and 6 is a divisor of 24 mean the same thing.)
5 divides 0 because . In fact, every integer except zero divides zero.
7 is a divisor of 49 since .
7 divides 7 since .
1 divides 5 because . It is, however, not a proper divisor.
-3 divides 9 because
-4 divides -16 because
2 does not divide 9 because there is no integer k such that . 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 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 is a divisor of a (we also say that d divides , this fact may be expressed by writing . Similarly, if does not divide , we write . For example, but .

Related Concepts

(If is a divisor of (), we say is a multiple of . For example, since , 12 is a multiple of 4. If both and , we say is a common multiple of and . Ignoring the sign (i.e., only considering nonnegative integers), there is a unique greatest common divisor of any two integers and written or, more commonly, . 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 and is the smallest (positive) integer such that and . Thus, the least common multiple of 12 and 9 is 36 (written ).

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.