Does $p^k$ divide ${}_n\!C_r$ for all integer r if $p^k|n$ where $0\leq r \leq n$ and $p$ is prime?
Does $k$-th power of $p$ divide ${}_n\!C_r$ if the previous divides $n$?
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elementary-number-theory
binomial-coefficients
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0I think it's $\binom{n}{r}$ (the number of $k$-combinations of $n$/the binomial coefficient). – 2012-07-05
1 Answers
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The answer is no. Take $p=3$, $n=6$, $k=1$, $r=3$. Then $p^k\mid n$ , but $p^k\nmid{}_n C_r$
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0Or $2^2\nmid{4\choose2}$. Or pretty much anything with $r=0$ or with $r=n$. – 2012-07-06