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    DaphneV's Avatar
    DaphneV Posts: 1, Reputation: 1
    New Member
     
    #1

    Dec 10, 2016, 11:06 AM
    Find the number of divisors of 189,720 that are composite numbers?
    So, this question comes in two parts:

    Find the number of divisors of 189,720 that are (a) composite numbers, and (b) divisible by five.

    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 we used in class, but I'd appreciate if someone could look over this and tell if it was right! Thank you!
    ebaines's Avatar
    ebaines Posts: 12,131, Reputation: 1307
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    #2

    Dec 14, 2016, 06:58 AM
    How did yui decide to use 4, 6, and 15 as divisors? I don't know the "tree" method, but I would think that you need to determine the prime factors of 189720, and then use (p+1)(q+1)r+1)(s+1)... where p. q. r. s. .. are the quantity of each prime factor. Let me give an example: consider the number 72. Its prime factors are 2,2,2,3,3; so we use (p+1)(q+1) = 4*3 = 12. Thus there are twelve divisors of 72. Two of those divisors are prime (2 and 3), and one of those is the number 1 and one of those is 72 itself, so that means 12-4 = 8 divisors of 72 are composite numbers. They are: 4, 6, 8, 9, 12, 18, 24, and 36. I suggest you follow this same procedure for 189720.

    **EDIT** OK, I think I know what the tree method is, and you used it incorrectly. Your first line is OK: 4 x 47430 = 189720. But the second line is incorrect, it should include both 2 x 2 = 4 and 6 x 7905 = 47430. Then the third line would be 2x3 = 6 and 15 x 527 = 7905. Then keep going: 527 is divisible by 17. If you complete the tree it would look like the attached. Note that there are 3 instances of prime number 2, two instances of prime number 3, and one each of prime numbers 5, 17 and 31.

    Name:  Tree Method.JPG
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