Joe,
There is a nice way to determine the number of divisors of a positive integer and I am going to illustrate it with a smaller number, 360.
The first step is to write the prime factorization of 360
360 = 2^{3} 3^{2} 5^{1}
We are looking for divisors of 360 so suppose that n divides 360. Notice that if an integer divides n it also divides 360 so, in particular, if a prime divides n then that prime must be 2, 3 or 5. Hence n can be written
n = 2^{r} 3^{s} 5^{t}
You have to be a little careful here since, for example 36 divides 360 but
36 = 2^{2} 3^{2}
so I use the convention that a^{0} = 1 and write
36 = 2^{2} 3^{2} 5^{0}
What are the possible values of r, s and t? r can be 0, 1, 2 or 3. (r can't be 4 since 2^{4} does not divide 360.) likewise s can be 0, 1 or 2 and t can be 0 or 1. Hence if you are going to construct a divisor n of 360 you can form
n = 2^{r} 3^{s} 5^{t}
with 4 choices for r (0, 1, 2 or 3), 3 choices for s (0, 1 or 2) and 2 choices for t (0 or 1), hence there are
4 3 2 = 24
ways you can form n and thus 360 has 24 divisors.
Thus to calculate the number of divisors of 360
 write the prime factorization of 360 [360 = 2^{3} 3^{2} 5^{1}]
 look at the powers in the prime factorization [3, 2, 1]
 add 1 to each power [(3 + 1), (2 + 1), (1 + 1)]
 find the product of these numbers
[4 3 2 = 24]
Check this method with another number like 140. It's prime factorization is
2 x 2 x 5 x 7.
So that should be 3 choice x 2 choices x 2 choices = 12 divisors. Here they are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140.
Steve and Penny
