Number of the divisors: Difference between revisions

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~~The '''number~~ of divisors~~''' of a number (<tt>''d(n)''</tt>) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.~~

* If the number ~~<math>n</math> has the [[prime factorization]]~~

The '''number of divisors''' {{w|Divisor function|d(''n'')}} of a number ''n'' can be found from its [[prime factorization]] as the product of the by-one incremented exponents of all present prime bases.

~~:: <math>n = p_1^~~{~~e_1}\cdot p_2^~~{~~e_2~~}~~\dotsm p_r^{e_r~~}~~,</math>~~

~~: then:<ref>G. H. Hardy, E. M. Wright:~~ ''~~An Introduction to the Theory of Numbers.~~'' ~~4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.</ref>~~

~~:: <math>d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)</math>~~

* for [[~~coprime~~]] ~~numbers <math>m</math> and <math>n</math> applies:~~

~~:: <math>d(mn) = d(m)\cdot d(n)</math>~~

== ~~Some~~ Examples ==

If the number ''n'' has the prime factorization

<math>\displaystyle n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>

then:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers.'' 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.</ref>

<math>\displaystyle d(n) = (e_1 + 1)(e_2 + 1) \dotsm (e_r + 1)</math>

For {{w|Coprime integers|coprime}} numbers ''m'' and ''n'' it follows that

<math>\displaystyle d(mn) = d(m) \cdot d(n)</math>

== Examples ==

{| class="wikitable"

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== External links ==

* [https://en.wikipedia.org/wiki/Table_of_divisors Table of divisors - Wikipedia]

* [https://en.wikipedia.org/wiki/Divisor_function Divisor function - Wikipedia]

* {{OEIS|A000005}}

[[Category:Math]]

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-The '''number of divisors''' of a number (<tt>''d(n)''</tt>) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.
+{{Wikipedia| Table of divisors }}
-* If the number <math>n</math> has the [[prime factorization]]
+The '''number of divisors''' {{w|Divisor function|d(''n'')}} of a number ''n'' can be found from its [[prime factorization]] as the product of the by-one incremented exponents of all present prime bases.
-:: <math>n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>
-: then:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers.'' 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme&nbsp;273, p.&nbsp;239.</ref>
-:: <math>d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)</math>
-* for [[coprime]] numbers <math>m</math> and <math>n</math> applies:
-:: <math>d(mn) = d(m)\cdot d(n)</math>
-== Some Examples ==
+If the number ''n'' has the prime factorization
+<math>\displaystyle n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>
+then:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers.'' 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme&nbsp;273, p.&nbsp;239.</ref>
+<math>\displaystyle d(n) = (e_1 + 1)(e_2 + 1) \dotsm (e_r + 1)</math>
+For {{w|Coprime integers|coprime}} numbers ''m'' and ''n'' it follows that
+<math>\displaystyle d(mn) = d(m) \cdot d(n)</math>
+== Examples ==
 {| class="wikitable"
@@ Line 38: / Line 45: @@
 == External links ==
-* [https://en.wikipedia.org/wiki/Table_of_divisors Table of divisors &#45; Wikipedia]
-* [https://en.wikipedia.org/wiki/Divisor_function Divisor function &#45; Wikipedia]
 * {{OEIS|A000005}}
 [[Category:Math]]