Fact: The number 12 has 6 factors: 1, 2, 3, 4, 6, and 12.
Fact: The number 15 has 4 factors: 1, 3, 5, and 15.
But what about a number like, say, 1440? Is there a way to count the number of factors it has, without actually listing all the factors?
Fortunately, there is. First, find the prime factorization of the number. In the case of 1440, that's:
25 * 32 * 5.
Now, take the exponents of each prime factor: 5, 2, and 1. Add 1 to each of the exponents, to get 6, 3, and 2. Now multiply these values together:
6 * 3 * 2 = 36 factors.
In other words, if the exponents in the prime factorization are a1, a2, a3, ..., an, then the total number of factors is:
(a1+1)(a2+1)(a3+1)...(an+1).
Alas, I am too lazy to explain why this formula works. But if you really want to know, list all the factors of a number, write the prime factorization of each factor, and look for a pattern.