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Question Number 167110 by nimnim last updated on 06/Mar/22

  Find the remainder when:−    (a)  41! is divided by 1681    (b) 225! is divided by 227    (c) 15! is divided by 19

$$\:\:{Find}\:{the}\:{remainder}\:{when}:− \\ $$$$\:\:\left({a}\right)\:\:\mathrm{41}!\:{is}\:{divided}\:{by}\:\mathrm{1681} \\ $$$$\:\:\left({b}\right)\:\mathrm{225}!\:{is}\:{divided}\:{by}\:\mathrm{227} \\ $$$$\:\:\left({c}\right)\:\mathrm{15}!\:{is}\:{divided}\:{by}\:\mathrm{19} \\ $$

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]

$$\begin{array}{|c|}{{p}\in\mathbb{P}\Leftrightarrow\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right]}\\\hline\end{array}\: \\ $$$$\left({b}\right) \\ $$$$\left(\mathrm{227}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[\mathrm{227}\right]\:\:\:\:\:\:\left[\because\mathrm{227}\in\mathbb{P}\right] \\ $$$$\mathrm{226}!\equiv−\mathrm{1}\left[\mathrm{227}\right] \\ $$$$\mathrm{226}!\equiv−\mathrm{1}+\mathrm{227}\left[\mathrm{227}\right] \\ $$$$\mathrm{226}!\equiv\mathrm{226}\left[\mathrm{227}\right] \\ $$$$\mathrm{225}!\equiv\mathrm{1}\left[\mathrm{227}\right] \\ $$$$\left({c}\right) \\ $$$$\left(\mathrm{19}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[\mathrm{19}\right]\:\:\:\:\:\left[\because\:\mathrm{19}\in\mathbb{P}\right] \\ $$$$\mathrm{18}!\equiv−\mathrm{1}+\mathrm{19}\left[\mathrm{19}\right] \\ $$$$\mathrm{18}!\equiv\mathrm{18}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{1}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{1}+\mathrm{19}×\mathrm{8}\left[\mathrm{19}\right] \\ $$$$\mathrm{17}!\equiv\mathrm{153}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{9}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{9}+\mathrm{19}×\mathrm{13}\left[\mathrm{19}\right] \\ $$$$\mathrm{16}!\equiv\mathrm{256}\left[\mathrm{19}\right] \\ $$$$\mathrm{15}!\equiv\mathrm{16}\left[\mathrm{19}\right] \\ $$

Commented by nimnim last updated on 06/Mar/22

Nice..  Thank you Sir..

$${Nice}.. \\ $$$${Thank}\:{you}\:{Sir}.. \\ $$

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