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December 10th, 2016, 09:15 AM   #1
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Find the number of divisors of 189,720 that are composite numbers?

I have question from my Data Management class that asks for the number of divisors of 189,720 that are composite numbers.

I tried to solve the question by using a tree diagram, as I was shown in class.

(a)

189,720
4 x 47,430
6 x 7,605
15 x 37, 944

(Sorry if the diagram isn't as clear online, but it's shaped like a tree!)

Then, I did this:

(p+1)(q+1)(r+1)
= (4+1)(6+1)(15+1)
= 560 divisors

(b)

189,720
= 5 x 37,944

(p+1)(q+1)(r+1)
= (5+1)
= 6

I tried to follow the method used in class, but the example didn't specifically ask for composite numbers, so I'm not entirely sure if this is right. I'd appreciate if someone could look over this and tell if it was correct! Thank you!
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December 10th, 2016, 10:34 AM   #2
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well... I can't comment on the method used as I'm not familiar with it but conceptually what you want to do is

a) create a list of the prime factors of 189720. If a factor has degree $n$ then repeat that factor $n$ times. For example $36 \to (2,2,3,3)$

b) take all the unordered subsets of this list and find the product of the elements of that subset. Remove any duplicates.

c) remove the primes from the set of numbers found in (b)

d) count the number of elements remaining after (c)

is there some tree based combinatorial magic so that you don't have to go through all the above? apparently. Maybe the above will help you to remember what they were doing in class.

I get 91 composite numbers.
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December 11th, 2016, 07:08 PM   #3
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Thanks for the response! I managed to get 91 composite numbers through my method like this:

(2^3) x (3^2) x 5 x 17 x 31

Then, I used the formula: (p+1)(q+1)(r+1)...-1

= 4 x 3 x 2 x 2 x 2
= 96

96 - (prime numbers)
= 96 - 5
= 91

The second part of the question, which I now realize that I didn't include above, asks to find the number of divisors of 189,720 that are divisible by 5. If I use the same method to solve this question, I also get the answer 91...

96 (total number of divisors found) - 5 (divisors not divisible by 5) = 91

For some reason, I feel like I'm missing something about this second part. Can anyone confirm if this answer is correct, or if there is another method I should be using to solve this part of the question?
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