How many divisors does the number 138600 have? And is there a specific way to figure this out?

 

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 = 23 32 51

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 = 2r 3s 5t

You have to be a little careful here since, for example 36 divides 360 but

36 = 22 32

so I use the convention that a0 = 1 and write

36 = 22 32 50

What are the possible values of r, s and t? r can be 0, 1, 2 or 3. (r can't be 4 since 24 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 = 2r 3s 5t

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 = 23 32 51]
- 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