Prove that if $p$ is an odd prime and $k$ is an integer such that $1\leq k\leq p-1$, then,
$\binom{p-1}{k}\equiv (-1)^{k}(mod p)$
Binomial Residue
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Phlembac Adib Hasan
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Re: Binomial Residue
এইটা আমার সেলফ-মেড। সোয়া দুই বছর আগের।
http://www.matholympiad.org.bd/forum/vi ... 7433#p7433
http://www.matholympiad.org.bd/forum/vi ... 7433#p7433
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Re: Binomial Residue
Actually I got it from Telang's "Number Theory", chapter $5$ exercise.Phlembac Adib Hasan wrote:এইটা আমার সেলফ-মেড। সোয়া দুই বছর আগের।
http://www.matholympiad.org.bd/forum/vi ... 7433#p7433
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$
Re: Binomial Residue
SANZEED wrote:Actually I got it from Telang's "Number Theory", chapter $5$ exercise.Phlembac Adib Hasan wrote:এইটা আমার সেলফ-মেড। সোয়া দুই বছর আগের।
http://www.matholympiad.org.bd/forum/vi ... 7433#p7433
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sakibtanvir
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Re: Binomial Residue
$(p-1)!\equiv(p-1)(p-2)(p-3)...(p-k)(p-k-1)!\equiv(-1)^{k}k!(p-k-1)!(modp)$
As p is odd prime,$gcd(p,k!(p-k-1)!)=1$.So we can divide both side of the equation by $k!(p-k-1)!$.
As p is odd prime,$gcd(p,k!(p-k-1)!)=1$.So we can divide both side of the equation by $k!(p-k-1)!$.
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