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Math Help - proof

  1. #1
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    proof

    Start with a finite sequence a1, a2, . . . , an of positive
    integers. If possible, choose two indices j < k such
    that aj does not divide ak, and replace aj and ak by
    gcd(aj , ak) and lcm(aj , ak), respectively. Prove that if
    this process is repeated, it must eventually stop and the
    final sequence does not depend on the choices made.
    (Note: gcd means greatest common divisor and lcm
    means least common multiple.)
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  2. #2
    MHF Contributor

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    Paris, France
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    Quote Originally Posted by RiemannManifold View Post
    Start with a finite sequence a1, a2, . . . , an of positive
    integers. If possible, choose two indices j < k such
    that aj does not divide ak, and replace aj and ak by
    gcd(aj , ak) and lcm(aj , ak), respectively. Prove that if
    this process is repeated, it must eventually stop and the
    final sequence does not depend on the choices made.
    (Note: gcd means greatest common divisor and lcm
    means least common multiple.)
    What do you think about it? Did you try with a few numbers, what did you notice? Do you have a conjecture, a first draft you would submit to us?
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