Counting divisors
The general task of counting divisors of any integer requires us to organize the divisors of an integer. The prime factorization of the integer gives us a way to describe, and therefore organize, these divisors.
Example: 72
Consider the task of counting the divisors of 72.
Since each divisor of 72 can have a power of 2, and since this power can be 0, 1, 2, or 3, we have 4 possibilities. Likewise, since each divisor can have a power of 3, and since this power can be 0, 1, or 2, we have 3 possibilities. By an elementary counting principle, we have 3*4=12 divisors.
Formula
Generally, If we have a number's prime factorization, the number of divisors is equal to the product of each of the exponents plus one, i.e. 
where each of the 
are the exponents of the nth unique exponentiation base.
