There is a little nice

theorem which allows us to find how many

divisors
a

number has, if we know its

Prime Factorization.

Consider an

integer number,

**n**.

Suppose

**n** = (p_{1})^{a1}
* (p_{2})^{a2} * ... * (p_{k})^{ak}
,where

p_{1}...

p_{k} are

prime numbers,
and

a_{1}...

a_{k} are the corresponding

exponents.

Then, you can easily find out how many divisors n has (say S), with the following
simple

formula:

S = (

a_{1}+1)*(

a_{2}+1)*...*(

a_{k}+1)

*Proof*
Suppose as above **n** = (p_{1})^{a1}
* (p_{2})^{a2} * ... * (p_{k})^{ak}

The divisors of the number **n** then are those with prime factorizations with the same primes as **n** but with powers no bigger than the powers a_{i}. Each power can be chosen independently, so there are (a_{1}+1)(a_{2}+1)...(a_{k}+1) such divisors.

Let's see that through an

example. Consider 24.

24 = 2

^{3} * 3

^{1}
So, with the above formula we take (3+1)*(1+1) = 8 divisors

Let's check this.

Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24

OK, they are 8.

Source: The Papyrous, Larousse, Britannica Encyclopedia