I know that one way to find the number of divisors is to find the prime factors of that number and add one to all of the powers and then multiply them together so for example
$$555 = 3^1 \cdot 5^1 \cdot 37^1$$
therefore the number of divisors = $(1+1)(1+1)(1+1) = 2 \cdot 2 \cdot 2 = 8$
What I do not know (and can't seem to find when I look) is WHY this relationship exists or a formula which shows its proof.
Has anyone seen such a formula?