### Counting Divisors

Sometimes you need to know the number of divisors a number has. For small numbers it is possible to list all the factors and count them. For example by listing the factor pairs for 24 you can see that there are 8 divisors.

 24 = 1 · 24 = 2 · 12 = 3 · 8 = 4 · 6

What if you need to know how many divisors a large number has. It is impractical to list all the factor pairs if there are hundreds of divisors. There is an easier way to count factors.

Notice that all divisors of a number must be made up of prime factors of that number. If you start by prime factoring the number, then the divisor count is the number of possible combinations of the prime factors.

How many divisors does 126,000 have?

Start by prime factoring:

 126000 = 24 · 32 · 53 · 7

How many ways can the prime factors be combined? 2 can be present with one of five different exponents: 0, 1, 2, 3, or 4. Similarly, 3 can have exponents 0-2, 5 can have exponents 0-3, and 7 can have exponents 0-1. The number of possible combinations is therefore

 5 · 3 · 4 · 2 = 120

 126,000 has 120 divisors.

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