Number of the divisors: Difference between revisions
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{{Wikipedia| Table of divisors }} | |||
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. | |||
== | 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" | {| class="wikitable" | ||
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== External links == | == External links == | ||
* {{OEIS|A000005}} | * {{OEIS|A000005}} | ||
[[Category:Math]] | [[Category:Math]] | ||
Latest revision as of 15:13, 19 October 2023
The number of divisors 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.
If the number n has the prime factorization
[math]\displaystyle{ \displaystyle n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r}, }[/math]
then:[1]
[math]\displaystyle{ \displaystyle d(n) = (e_1 + 1)(e_2 + 1) \dotsm (e_r + 1) }[/math]
For coprime numbers m and n it follows that
[math]\displaystyle{ \displaystyle d(mn) = d(m) \cdot d(n) }[/math]
Examples
| Number | Prime factorization | Number of divisors |
|---|---|---|
| 8 | [math]\displaystyle{ 8 = 2^3 }[/math] | [math]\displaystyle{ d(8) = (3+1) = 4 }[/math] |
| 12 | [math]\displaystyle{ 12 = 2^2 \cdot 3 }[/math] | [math]\displaystyle{ d(12) = (2+1)(1+1) = 6 }[/math] |
| 30 | [math]\displaystyle{ 30 = 2 \cdot 3 \cdot 5 }[/math] | [math]\displaystyle{ d(30) = (1+1)(1+1)(1+1) = 8 }[/math] |
See also
References
- ↑ 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.