Number of the divisors
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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.