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Math Help - Divisibility problem

  1. #1
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    Divisibility problem

    If a, b, c are different positive integers such that a is divisible by b,
    and b is divisible by c, which of the following statements must be true?
    I. a is divisible by c
    II. a has at least 3 positive factors
    III. a = bc

    How can I prove without using numbers that only I and II are valid?
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  2. #2
    Member Goku's Avatar
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    Re: Divisibility problem

    b=ck for some integer k, [1]

    a=bn for some integer n. [2]

    substitute [1] into [2]:

    you get a = ckn

    since kn is an integer:

    you have a is divisble by c.
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  3. #3
    MHF Contributor

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    Re: Divisibility problem

    II depends on what you allow to be a factor. if a is allowed as a factor, then yes: a, b and c are obviously factors of a (by Goku's answer above).

    if a factor must be "smaller than a", then no: consider a = 4, b = 2, c = 1.

    to "disprove" III, it suffices to give a counter-example: let a = 12, b = 6, c = 3. then 12 is divisible by 6, and 6 is divisible by 3, but 12 is not 3*6.
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