Question Number 167110 by nimnim last updated on 06/Mar/22
$$\:\:{Find}\:{the}\:{remainder}\:{when}:− \\ $$$$\:\:\left({a}\right)\:\:\mathrm{41}!\:{is}\:{divided}\:{by}\:\mathrm{1681} \\ $$$$\:\:\left({b}\right)\:\mathrm{225}!\:{is}\:{divided}\:{by}\:\mathrm{227} \\ $$$$\:\:\left({c}\right)\:\mathrm{15}!\:{is}\:{divided}\:{by}\:\mathrm{19} \\ $$
Answered by Rasheed.Sindhi last updated on 06/Mar/22
![determinant (((p∈P⇔(p−1)!≡−1[p]))) (b) (227−1)!≡−1[227] [∵227∈P] 226!≡−1[227] 226!≡−1+227[227] 226!≡226[227] 225!≡1[227] (c) (19−1)!≡−1[19] [∵ 19∈P] 18!≡−1+19[19] 18!≡18[19] 17!≡1[19] 17!≡1+19×8[19] 17!≡153[19] 16!≡9[19] 16!≡9+19×13[19] 16!≡256[19] 15!≡16[19]](https://www.tinkutara.com/question/Q167111.png)
$$\begin{array}{|c|}{{p}\in\mathbb{P}\Leftrightarrow\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right]}\\\hline\end{array}\: \\ $$$$\left({b}\right) \\ $$$$\left(\mathrm{227}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[\mathrm{227}\right]\:\:\:\:\:\:\left[\because\mathrm{227}\in\mathbb{P}\right] \\ $$$$\mathrm{226}!\equiv−\mathrm{1}\left[\mathrm{227}\right] \\ $$$$\mathrm{226}!\equiv−\mathrm{1}+\mathrm{227}\left[\mathrm{227}\right] \\ $$$$\mathrm{226}!\equiv\mathrm{226}\left[\mathrm{227}\right] \\ $$$$\mathrm{225}!\equiv\mathrm{1}\left[\mathrm{227}\right] \\ $$$$\left({c}\right) \\ $$$$\left(\mathrm{19}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[\mathrm{19}\right]\:\:\:\:\:\left[\because\:\mathrm{19}\in\mathbb{P}\right] \\ $$$$\mathrm{18}!\equiv−\mathrm{1}+\mathrm{19}\left[\mathrm{19}\right] \\ $$$$\mathrm{18}!\equiv\mathrm{18}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{1}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{1}+\mathrm{19}×\mathrm{8}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{153}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{9}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{9}+\mathrm{19}×\mathrm{13}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{256}\left[\mathrm{19}\right] \\ $$$$\mathrm{15}!\equiv\mathrm{16}\left[\mathrm{19}\right] \\ $$
Commented by nimnim last updated on 06/Mar/22
$${Nice}.. \\ $$$${Thank}\:{you}\:{Sir}.. \\ $$