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Math Help - Divisibility proof in random integer subset

  1. #1
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    Divisibility proof in random integer subset

    Can anyone help me getting started on this one? I figure it has something to do with the pigeonhole principle, but I don't know where to start.

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  2. #2
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    Quote Originally Posted by knarQ View Post
    Can anyone help me getting started on this one? I figure it has something to do with the pigeonhole principle, but I don't know where to start.

    We can prove a more general result. If A is a subset with n+1 elements from \{1,2,3,...,2n\} then there exists a,b\in A with a|b. Each element in \{1,2,...,2n\} can be written as n\cdot 2^m where n is odd and m\geq 0. Let these be our pigeonholes. We see that there are n pigeonholes because there are n odd numbers between 1,2,...,2n. Thus, two numbers end up in the same pigeonhole. So a=n\cdot 2^{m_1}, b=n\cdot 2^{m_2} not it is clear that a|b if m_1<m_2.
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  3. #3
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    It took me a while to get this but it's a really nice proof. I was thinking in the lines of odd numbers and multiples of odd numbers but i couldn't get it down on paper, so thank you very much!
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