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Math Help - Neckles and diamods

  1. #1
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    Neckles and diamods

    There are at least 2 necklaces. On each necklace there is the same number of diamonds, on each at least 2. If we knew the number of all diamonds, we would also know the number of necklaces. The number of all diamonds is bigger than 200 and smaller than 300. How many necklaces is there?

    (A) 16
    (B) 17
    (C) 19
    (D) 25
    (E) Some other number
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  2. #2
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    Hello, metlx!

    There are at least 2 necklaces.
    On each necklace there is the same number of diamonds, on each at least 2.
    If we knew the number of all diamonds, we would also know the number of necklaces.
    The number of all diamonds is bigger than 200 and smaller than 300.
    How many necklaces is there?

    . . (A)\;16 \qquad(B)\;17\qquad(C)\;19 \qquad(D) \;25 \qquad (E)\text{ other}

    There is no formula for this problem.
    It is an exercise in reasoning and logic.


    Let n = number of necklaces.
    Let d = number of diamonds on each necklace.

    So there is a total of: . D \,=\,nd diamonds . . . and: 200 < D < 300

    And we are told: if we know D, we'd know the number of necklaces.
    . . [A powerful clue!]


    Suppose D has two different prime factors.
    . . For example: . D\,=\,203 \,=\,7\cdot29

    Then we could have two scenarios:
    . . 7 necklaces with 29 diamonds on each.
    . . 29 necklaces with 7 diamonds on each.


    The only way we'd be sure of the number of necklaces
    . . is if the two factors are equal.

    The only square of a prime number in that range is: . 17^2 \,=\,289

    Therefore, there are 17 necklaces with 17 diamonds on each . . . answer (B).

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