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Question Number 62452 by Tawa1 last updated on 21/Jun/19

Find the remainder when 2014! is divisible by 2017

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\:\:\mathrm{2014}!\:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\:\mathrm{2017} \\ $$

Answered by Rasheed.Sindhi last updated on 21/Jun/19

Wilson′s Theorm: (p−1)!≡−1(mod p) : p∈P ∵ 2017∈P ∴ (2017−1)!≡−1(mod 2017) 2016!≡−1(mod 2017) 2016!≡−1+2017=2016(mod 2017) 2015!≡1(mod 2017) 2(2015!)≡2(mod 2017) 2(2015!)≡2−2017=−2015(mod 2017) 2(2014!)≡−1(mod 2017) 2(2014!)≡−1+2017=2016(mod 2017) 2014!≡2016/2(mod 2017) 2014!≡1008(mod 2017) Remainder 1008

$${Wilson}'{s}\:{Theorm}: \\ $$$$\:\:\:\:\:\:\:\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left({mod}\:{p}\right)\::\:{p}\in\mathbb{P} \\ $$$$\:\:\:\because\:\:\:\:\mathrm{2017}\in\mathbb{P} \\ $$$$\:\:\:\:\therefore\:\left(\mathrm{2017}−\mathrm{1}\right)!\equiv−\mathrm{1}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2016}!\equiv−\mathrm{1}\left({mod}\:\mathrm{2017}\right)\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2016}!\equiv−\mathrm{1}+\mathrm{2017}=\mathrm{2016}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2015}!\equiv\mathrm{1}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{2015}!\right)\equiv\mathrm{2}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{2015}!\right)\equiv\mathrm{2}−\mathrm{2017}=−\mathrm{2015}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{2014}!\right)\equiv−\mathrm{1}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left(\mathrm{2014}!\right)\equiv−\mathrm{1}+\mathrm{2017}=\mathrm{2016}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2014}!\equiv\mathrm{2016}/\mathrm{2}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{2014}!\equiv\mathrm{1008}\left({mod}\:\mathrm{2017}\right) \\ $$$$\:\:\:\:\:\:{Remainder}\:\mathrm{1008} \\ $$

Commented by Tawa1 last updated on 21/Jun/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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