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Math Help - Positive divisors

  1. #1
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    Positive divisors

    There are x number of positive divisors of 72, y number of positive divisors of 900.

    The positive divisors of 900 that are not divisors of 72 are z.

    Hoping there's a formula and you're not actually supposed to go over all possible numbers...

    Any help appreciated.
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  2. #2
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    Quote Originally Posted by davidman View Post
    There are x number of positive divisors of 72, y number of positive divisors of 900.
    The positive divisors of 900 that are not divisors of 72 are z.
    I find this question a bit confusing. Is this what you mean?
    Suppose that t is the number of divisors of 72~\&~900.
    Then z=y-t. Note that t\le x.

    If that is not your question, please clarify in detail.
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  3. #3
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    Not quite sure what your t represents.

    z is the number of divisors of 900 minus the divisors that overlap with 72.
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  4. #4
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    Quote Originally Posted by davidman View Post
    Not quite sure what your t represents.
    z is the number of divisors of 900 minus the divisors that overlap with 72.
    What is the point of this reply?
    You did not read my reply? It says clearly what t equals.
    Do you have difficulty translating Enghish?
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  5. #5
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    Quote Originally Posted by Plato View Post
    What is the point of this reply?
    First, to tell you that I was not quite sure what you meant with what you defined t to be. In other words, 72~\&~900 is confusing and threw me off, so I do not understand what you mean when you say "the number of divisors of 72~\&~900". Is it the number of divisors that overlap for the two? Is it the sum of the number of divisors, not taking into regard the case of overlap?

    Second, to clarify what it was I meant. You did ask me to, didn't you?

    You did not read my reply? It says clearly what t equals.
    Do you have difficulty translating Enghish?
    English is my first language. I just don't have a very wide vocabulary when it comes to mathematics.

    But sure, I'll look over your post again for hints to what I might not be seeing clearly.
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  6. #6
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    What does it mean to say that a number is a divisor of seventy-two and ninety?
    Does it mean that the number divides 72 and divides 90?
    BTW. ‘Overlap’ in not mathematical.
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  7. #7
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    Quote Originally Posted by Plato View Post
    What does it mean to say that a number is a divisor of seventy-two and ninety?
    Does it mean that the number divides 72 and divides 90?
    Yes, that makes sense. If we assume that is what it means, how do you figure out the number of divisors of a certain number (or two numbers for that matter)? Would be great to know how.
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  8. #8
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    Any positive integer can be factors as powers of primes.
    900=2^2\cdot 3^2\cdot 5^2 look at the exponents:
    Then add one to each exponent and multiply: (2+1)(2+1)(2+1)=27.
    So there are 27 divisors of 900.

    72=2^3\cdot 3^2, so (3+1)(2+1)=12.
    There 12 divisors of 72.

    The greatest common divisor: \text{GCD}(900,72)=2^2\cdot 3^2 so (2+1)(2+1)=9 common divisors of both 72 and 900.

    Thus there are 27-9=18 divisors of 900 that are not divisors of 72.
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  9. #9
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    Thank you so much! It was a lot more straightforward than I expected.
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