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Thread: Divisibility of binomial coefficients, countability of N^k.

  1. #1
    LHS
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    Exclamation Divisibility of binomial coefficients, countability of N^k.

    If anyone could shed some light on this problem, especially the second part to part a, I would be most grateful!

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  2. #2
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    Quote Originally Posted by LHS View Post
    If anyone could shed some light on this problem, especially the second part to part a, I would be most grateful!



    Hmmm...what's the question? How to take the frog out without first melting the ice?

    Tonio
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  3. #3
    LHS
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    Sorry? I don't understand, can you not see the inbeded image?

    If not, here it is
    http://img217.imageshack.us/img217/921/48698386.jpg
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  4. #4
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    Quote Originally Posted by LHS View Post
    Sorry? I don't understand, can you not see the inbeded image?

    If not, here it is
    http://img217.imageshack.us/img217/921/48698386.jpg

    I'm afraid somebody's hacked the MHF: all over the place appear those annoying frogs inside ice cubes, and

    that's what I saw instead your embedded image.

    Anyway, following the link above I get...a frog inside an ice cube again! I don't know what's going on.

    Perhaps somebody hacked the imageshack site and now we've been infected...

    Tonio
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  5. #5
    LHS
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    Haha.. right, ok, that certainly explains it! No worries!
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  6. #6
    A Plied Mathematician
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    Here's the question:

    (a) Let $\displaystyle p\in\mathbb{Z}$ be a prime number. Prove that for any natural number $\displaystyle r$ such that $\displaystyle 0<r<p$ the binomial coefficient $\displaystyle p\choose r$ is divisible by $\displaystyle p.$ Hence prove that, for any positive integer $\displaystyle n,$ the integer $\displaystyle n^{p}-n$ is divisible by $\displaystyle p.$

    (b) Let $\displaystyle k\in\mathbb{N}$ with $\displaystyle k\ge 2$ and let $\displaystyle \varphi:\mathbb{N}^{k}\to\mathbb{N}$ be given by

    $\displaystyle \varphi((a_{1},\dots,a_{k}))=2^{a_{1}+1}\,3^{a_{2} +1}\dots p_{k}^{a_{k}+1},$

    where $\displaystyle p_{k}$ is the $\displaystyle k$th prime. Deduce from the Fundamental Theorem of Arithmetic that $\displaystyle \varphi$ is injective and hence that $\displaystyle \mathbb{N}^{k}$ is a countable set for each $\displaystyle k\ge 2.$

    [EDIT]: Both links and image are fine for me.
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