Number of the divisors

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The number of divisors of a number (d(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:[1]
[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

Number Prime factorization Number of divisors
8 [math]8 = 2^3[/math] [math]d(8) = (3+1) = 4[/math]
12 [math]12 = 2^2 \cdot 3[/math] [math]d(12) = (2+1)(1+1) = 6[/math]
30 [math]30 = 2 \cdot 3 \cdot 5[/math] [math]d(30) = (1+1)(1+1)(1+1) = 8[/math]

References

  1. 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.

External links