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Math Help - composite positive integers

  1. #1
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    composite positive integers

    determine all composite positive integers n for which it is possible to arrange all divisors of n that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

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  2. #2
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    Quote Originally Posted by nh149 View Post
    determine all composite positive integers n for which it is possible to arrange all divisors of n that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

     n = p^k, k>2
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    A proof would be nice; I think that's what the problem is asking for.

    Edit : 9 3 6 12 2 4 18 36 is a counter-example to your "solution".
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    Quote Originally Posted by Bruno J. View Post
    A proof would be nice; I think that's what the problem is asking for.

    Edit : 9 3 6 12 2 4 18 36 is a counter-example to your "solution".
    Could you tell me what composite number you are using; that is, what prime you are raising to what power?
    .
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    MHF Contributor Bruno J.'s Avatar
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    Those are the divisors of 36. If you join them in a circle (so that 9 and 36 are adjacent), you will see that the criterium of the problem is satisfied, and 36 is not a prime-power.

    (2 4 6 3 12) is another counter-example, with the divisors of 12.
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  6. #6
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    Quote Originally Posted by Bruno J. View Post
    Those are the divisors of 36. If you join them in a circle (so that 9 and 36 are adjacent), you will see that the criterium of the problem is satisfied, and 36 is not a prime-power.

    (2 4 6 3 12) is another counter-example, with the divisors of 12.
    Sorry to be difficult, but I still cannot grasp how you can generate 12 or 36 from raising a prime number to a power.

    I'm at a loss here.

    Could you please show me how you can take a prime number and raise it to a power and come up with 12 or 36? Please.

    .
    I think I see what you are implying:
    BOTH p & k MUST be prime numbers:

     n = p^k
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  7. #7
    MHF Contributor Bruno J.'s Avatar
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    Are you feeling alright?
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  8. #8
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    Quote Originally Posted by Bruno J. View Post
    Are you feeling alright?
    I feel just fine.
    Thanks for asking.

    Could you show how it is possible to raise a prime number to any power and come up with 12 or 36?

    .
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    Quote Originally Posted by aidan View Post
    I feel just fine.
    Thanks for asking.

    Could you show how it is possible to raise a prime number to any power and come up with 12 or 36?

    .

    It is not possible. Why do you keep on asking? Is it that you meant in your answer p^k , k > 2, that p is a prime? And if you did why didn't you write so?
    And anyway: can you prove your claim?

    Tonio
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  10. #10
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    Quote Originally Posted by aidan View Post
    Sorry to be difficult, but I still cannot grasp how you can generate 12 or 36 from raising a prime number to a power.

    I'm at a loss here.

    Could you please show me how you can take a prime number and raise it to a power and come up with 12 or 36? Please.
    ? You are the only one who has said anything about raising a prime to a power. All that is being asked about here are composite positive integers. Bruno J's counter example was specifically to show that you do NOT have to have p^k.

    .
    I think I see what you are implying:
    BOTH p & k MUST be prime numbers:

     n = p^k
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