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Math Help - Multiple of p

  1. #1
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    Multiple of p

    Let . Show that is a multiple of for .

    How to do this question..?
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  2. #2
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    Quote Originally Posted by usagi_killer View Post
    Let . Show that is a multiple of for .

    How to do this question..?

    \binom{p^r}{k}=\frac{\left(p^r-(k-1)\right)\left(p^r-(k-2)\right)\cdot \ldots \cdot p^r}{1\cdot 2\cdot \ldots \cdot k} , so the solution follows at once from the following

    Lemma: For every 1\le j < k , we have that p^m\mid p^r-(k-j)\,\Longleftrightarrow\,p^m\mid k-j , for some m\in\mathbb{N}

    From the above, it turns out that the powers of p cancel each others in pairs (one from the numerator, one from the denominator), and p^r doesn't cancel COMPLETELY with no power of p from the denominator EXCEPT, perhaps, with k=ap^s...as k < p^r , it follows that  s<r and then \frac{p^r}{k}=\frac{p^{r-s}}{a} and we're done.

    Tonio
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  3. #3
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    lol sorry I don't really get what you mean.

    How did you prove they're multiples of p..?
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