Question

Find the remainder when \(40!\) is divided by \(43\)

(... follow up of previous problem http://openstudy.com/study#/updates/55028c14e4b06ef8a1430efb )

Answer 1

\(\large \color{black}{
\dfrac{40!}{43}\hspace{.33em}\\~\\
\implies \dfrac{42!\times\left(41\times 42\right)^{-1}}{43}\hspace{.33em}\\~\\
\implies \dfrac{(-1)\times\left(-2\times -1\right)^{-1}}{43}----------(\text{wilson theorem})\hspace{.33em}\\~\\
\implies \dfrac{(-1)\times\left(2\right)^{-1}}{43}\hspace{.33em}\\~\\
\implies \dfrac{(-1)\times22}{43}\hspace{.33em}\\~\\
\implies \dfrac{21}{43} \hspace{.33em}\\~\\
\Large \implies 21 \hspace{.33em}\\~\\
}\)

Answer 2

Neat :)