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.

$n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},$
then:[1]
$d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)$
• for coprime numbers $m$ and $n$ applies:
$d(mn) = d(m)\cdot d(n)$

Some Examples

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

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.