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.
===Further Reading===
*{{cite book
|last = Scharlau
|first = Winfried
|coauthors = Opolka, Hans
|title = From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development
|series = Undergraduate Texts in Mathematics
|publisher = Springer-Verlag
|date = 1985
|isbn = 0-387-90942-7 }}


*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.
[[Category:CZ Live]]
[[Category:CZ Live]]
[[Category:Mathematics Workgroup]]
[[Category:Mathematics Workgroup]]

Revision as of 18:39, 31 March 2007

Divisor (Number theory)

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.

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.