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Question Number 34140 by NECx last updated on 01/May/18
what is the remainder when 767^(1009) is divided by 25
$${what}\:{is}\:{the}\:{remainder}\:{when} \\ $$$$\mathrm{767}^{\mathrm{1009}} \:{is}\:{divided}\:{by}\:\mathrm{25} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/May/18
see 25×31=775 (775−8)^(1009) there is total 1009+1=1010 terms the terms consisting 775 are divisble by 25 now the last term that is(−8)^(1009 ) to be devided by 25 and remainder to find −(8).8^(1008 ) =−(8).(8^4 )^(252) =(−8).(4096)^(252)
$${see}\:\mathrm{25}×\mathrm{31}=\mathrm{775} \\ $$$$\left(\mathrm{775}−\mathrm{8}\right)^{\mathrm{1009}} \:{there}\:{is}\:{total}\:\mathrm{1009}+\mathrm{1}=\mathrm{1010}\:{terms} \\ $$$${the}\:{terms}\:{consisting}\:\mathrm{775}\:{are}\:{divisble}\:{by}\:\mathrm{25} \\ $$$${now}\:{the}\:{last}\:{term}\:{that}\:{is}\left(−\mathrm{8}\right)^{\mathrm{1009}\:\:} \\ $$$${to}\:{be}\:{devided}\:{by}\:\mathrm{25}\:{and}\:{remainder}\:{to}\:{find} \\ $$$$−\left(\mathrm{8}\right).\mathrm{8}^{\mathrm{1008}\:} =−\left(\mathrm{8}\right).\left(\mathrm{8}^{\mathrm{4}} \right)^{\mathrm{252}} \\ $$$$=\left(−\mathrm{8}\right).\left(\mathrm{4096}\right)^{\mathrm{252}} \:\:\: \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/May/18
=(−8)(4100−4)^(252) =f(25)+(−8)(−4)^(252) except last term (−8)(−4)^(252 ) remaining terms are divisble by 25 (−8)(4^4 )^(63) (−8)(250+6)^(63) terms consisting 250+(−8)(6)^(63) except(−8)(6)^(63 ) rest divisble by 25 (−8)(225−9)^(31) =terms containing 225+(−8)(−9)^(31) terms containing 225 divisble by 25 (8)(9)^(31) (8)(10−1)^(31) terms containing (8)(10^(31) −^(31) C1.10^(30) +^(31) C_2 .10^(29) so remainder is(8)(1)^(31) that is 8
$$=\left(−\mathrm{8}\right)\left(\mathrm{4100}−\mathrm{4}\right)^{\mathrm{252}} \\ $$$$={f}\left(\mathrm{25}\right)+\left(−\mathrm{8}\right)\left(−\mathrm{4}\right)^{\mathrm{252}} \\ $$$${except}\:{last}\:{term}\:\left(−\mathrm{8}\right)\left(−\mathrm{4}\right)^{\mathrm{252}\:} \:{remaining}\:{terms} \\ $$$${are}\:{divisble}\:{by}\:\mathrm{25} \\ $$$$\left(−\mathrm{8}\right)\left(\mathrm{4}^{\mathrm{4}} \right)^{\mathrm{63}} \\ $$$$\left(−\mathrm{8}\right)\left(\mathrm{250}+\mathrm{6}\right)^{\mathrm{63}} \\ $$$${terms}\:{consisting}\:\mathrm{250}+\left(−\mathrm{8}\right)\left(\mathrm{6}\right)^{\mathrm{63}} \\ $$$${except}\left(−\mathrm{8}\right)\left(\mathrm{6}\right)^{\mathrm{63}\:} {rest}\:{divisble}\:{by}\:\mathrm{25} \\ $$$$\left(−\mathrm{8}\right)\left(\mathrm{225}−\mathrm{9}\right)^{\mathrm{31}} \\ $$$$={terms}\:{containing}\:\mathrm{225}+\left(−\mathrm{8}\right)\left(−\mathrm{9}\right)^{\mathrm{31}} \\ $$$${terms}\:{containing}\:\mathrm{225}\:{divisble}\:{by}\:\mathrm{25} \\ $$$$\left(\mathrm{8}\right)\left(\mathrm{9}\right)^{\mathrm{31}} \\ $$$$\left(\mathrm{8}\right)\left(\mathrm{10}−\mathrm{1}\right)^{\mathrm{31}} \\ $$$${terms}\:{containing}\:\left(\mathrm{8}\right)\left(\mathrm{10}^{\mathrm{31}} −^{\mathrm{31}} {C}\mathrm{1}.\mathrm{10}^{\mathrm{30}} +^{\mathrm{31}} {C}_{\mathrm{2}} .\mathrm{10}^{\mathrm{29}} \right. \\ $$$${so}\:{remainder}\:{is}\left(\mathrm{8}\right)\left(\mathrm{1}\right)^{\mathrm{31}} \:{that}\:{is}\:\mathrm{8} \\ $$$$ \\ $$
Answered by Rasheed.Sindhi last updated on 04/May/18
767^5 ≡7(mod25) (767^5 )^2 ≡7^2 =49≡24≡−1(mod25) 767^(10) ≡−1(mod25) (767^(10) )^2 ≡(−1)^2 (mod 25) 767^(20) ≡1(mod 25) (767^(20) )^(50) ≡1(mod 25) 767^(1000) ≡1(mod 25)...................A 767^3 ≡13(mod 25) (767^3 )^3 ≡(13)^3 =2197≡22(mod 25) 767^9 ≡22(mod 25)........................B A×B: 767^(1009) ≡22(mod 25)
$$\mathrm{767}^{\mathrm{5}} \equiv\mathrm{7}\left(\mathrm{mod25}\right) \\ $$$$\left(\mathrm{767}^{\mathrm{5}} \right)^{\mathrm{2}} \equiv\mathrm{7}^{\mathrm{2}} =\mathrm{49}\equiv\mathrm{24}\equiv−\mathrm{1}\left(\mathrm{mod25}\right) \\ $$$$\mathrm{767}^{\mathrm{10}} \equiv−\mathrm{1}\left(\mathrm{mod25}\right) \\ $$$$\left(\mathrm{767}^{\mathrm{10}} \right)^{\mathrm{2}} \equiv\left(−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{mod}\:\mathrm{25}\right) \\ $$$$\mathrm{767}^{\mathrm{20}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{25}\right) \\ $$$$\left(\mathrm{767}^{\mathrm{20}} \right)^{\mathrm{50}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{25}\right) \\ $$$$\mathrm{767}^{\mathrm{1000}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{25}\right)……………….\mathrm{A} \\ $$$$\mathrm{767}^{\mathrm{3}} \equiv\mathrm{13}\left(\mathrm{mod}\:\mathrm{25}\right) \\ $$$$\left(\mathrm{767}^{\mathrm{3}} \right)^{\mathrm{3}} \equiv\left(\mathrm{13}\right)^{\mathrm{3}} =\mathrm{2197}\equiv\mathrm{22}\left(\mathrm{mod}\:\mathrm{25}\right) \\ $$$$\mathrm{767}^{\mathrm{9}} \equiv\mathrm{22}\left(\mathrm{mod}\:\mathrm{25}\right)……………………\mathrm{B} \\ $$$$\mathrm{A}×\mathrm{B}:\:\mathrm{767}^{\mathrm{1009}} \equiv\mathrm{22}\left(\mathrm{mod}\:\mathrm{25}\right) \\ $$
Commented by Rasheed.Sindhi last updated on 09/May/18
Mr NECx pl confirm that the answer is correct or not.
$$\mathrm{Mr}\:\mathrm{NECx}\:\mathrm{pl}\:\mathrm{confirm}\:\mathrm{that}\:\mathrm{the}\:\mathrm{answer} \\ $$$$\mathrm{is}\:\mathrm{correct}\:\mathrm{or}\:\mathrm{not}. \\ $$

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