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How can I find out last three digits

What is the last three digits of \(7^{9999}\)

Note by Fazla Rabbi
3 years, 9 months ago

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By Euler's Totient Theorem, \[7^{\phi(1000)}=7^{400}\equiv 1\pmod{1000}\] Now, \[7^{9999}=\dfrac{7^{10000}}{7}=\dfrac{(7^{400})^{25}}{7}\equiv\dfrac{1}{7}\pmod{1000}\] Now, all that remains is to find the inverse of 7 modulo 1000. We find that \[1001=7\cdot\boxed{143}\] Daniel Chiu · 3 years, 9 months ago

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@Daniel Chiu How do you find inverse of 7 modulo 1000? Nupur Prasad · 3 years, 9 months ago

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@Nupur Prasad Find \(x\) such that \(7x\equiv 1\pmod{1000}\). Daniel Chiu · 3 years, 9 months ago

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hello, we get 7^9999= 13655326938206299575388803133396753496736644114405381461829963962411828955569248281335981835273668310900484506718916901348651195294539829985150933969686285258869329236481285605531548183366701929103471910995171323770305208766008258258285092368934117908167663023499260087755337426770042841891448172308897121360310162326316854683467534489928627213854495698394965821277096168412307438025152950014352195174586595753311612597830605454382559444852642825933408175654807924020354818263710413323080455273492628015221339272913486928010852961275315658053876429989906252984220698056458599569321215985755913817620830138531191802515725941642192120168825337996051701530032549207981579956812952534204502646424494398101252430389082210190315474035528503219150993531780530279481987406986807840409864711145432654025574295844360237813572479788149581396753561271902967371574134570309187720836983617871988535658150641281664651310034419340732798919580952141063419330615759826815653168849669587208628030773458566503339210579765028956156317040136373383198150674629080794526161329125793027655429692168424480951702685391862625629437655471134887998949481311219148035668204991621779429552622067019596014680766323362940147777021112218367404441390348124973967196162618437237802803453804603913568420388378493913088862673315002440183918009799914011228448072321212925346915610549331756278370750059733337226900934442446540232632599709852529826465990963685940591737310263190900865992086797188005012451220721322602951050593881596632788059310063594466136965264630179180304509342668066327411764565787468819028067192689967007408479960830178465062006946653042492975519546811739635018985669909027634783454160151628253394950002427521000867513417041433448172829399054198433264630099214453164716552565679276670879204621719944403097797951470491745190322215294010715308151133429772510407526442158305553576969066789686913467547575899587837873307572542555457401200885316033878914540960524995195774675514703207152511047154948187496201043722610799187798125063981344190378543744924938438154273638755472904814705473141128186523613174667500664200023491113482049325479055237976183224273078944321168696318161662201188403103096215320017929654737692890216488519283299593533488759330719508105500564654264444722902309755009205890362443370026088725625404691859961086175156695246284103270143026088025747282475965657793617012763520224113888335151498131089125621277128488690418112302023633731098358233464019197011469107466714042251381664616076837293960648755558578310045314587020916497662639345475827466932253792980122408042511175724245674343322009979920432671103141325176818506304074014898273254328557495986865903003877458607924867895861284954381933894198991334641717544954540862655643624684820863842947012318661465665703445161942757923258343743090518489805300236705218214639997985538333169143476566628462348296847910693032266255826904278864804568069467700068214857493602909907968961853456788896313006228262293162499917971944303310372758175118534867652252680809334529150347518846173807297770704099159764751277963802133485748550600643310985468859064011536196975793176657844578241201335627872123486763592904006703005509521727187266652527173779603706912472770799729231598417471360003716878047628907063427754657800108418909505593341870072001014855884090685963051041609077531896465998474067172080961256475009566190873968666015457970355595511710147007940801533105221055039519718234650092237178964858828167487122877833678044854762480410677976454505764904885712954022721434096219716003365320211348765436324761007498533794277409291590087344811174193699068081056174900558440312968757170119018833846286737937917530321668518311971686956776924464038699522366685913158006568085870596586500154142201947232437000412492890513259592797655852609139010639863663333742527724929971952342622973254975319563136689761830710600850659669683156012471871389312164157127632682765533057324162523224417051019618122691001464491988346804462931386691417211834455003122234479832280470831503667053000595209701353071206567141187188105338244413300473076804890693948428036256402403914739227334417651479292046566019245969903336492800395251573701630351641659343653758501711893520231264953469166578100754139374150896987129005020272205964509625357981408678922421277109175504050979477011262603817567092329275271986388610180107866016775356719315747883323463246924669709835341382410045251567818279520565112141143418170392830084375756006580398505916662899978307355098423995194906094751321064992626623043216456861216424418176563790292172013787926691099978098942817126840254512616453653898985879013641509466389678987804351885069598225688345990845128007984813762967010494206906151160036869834166973057110496295157736901779280378756663756060716873234043100818455442651081491199637296753587252425709454882487539076107566142685430507075345682167510905105250475966275136266280454469464289465455165773290327611372413729633982470001751708727867868360464248586777378439172243929048551087211343643050571791167524143357689919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so our answer is 143

Sonnhard. Ryan Soedjak · 3 years, 9 months ago

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@Ryan Soedjak Wow, what an elegant solution! Michael Tong · 3 years, 9 months ago

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@Michael Tong hello, I think so too

Sonnhard. Ryan Soedjak · 3 years, 9 months ago

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@Michael Tong trolled :P Jun Das · 3 years, 9 months ago

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@Michael Tong I agree Aiman Rafeed · 3 years, 9 months ago

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@Ryan Soedjak Amazing solution. Nice thought process. :p Dhruv Bhasin · 3 years, 9 months ago

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@Ryan Soedjak Your profile pic and message see to have an effect on your posts... Yash Talekar · 3 years, 9 months ago

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@Yash Talekar actually it's my posts that affects my avatar. Ryan Soedjak · 3 years, 9 months ago

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@Ryan Soedjak Ugh,you took up 2 extra bytes from my cache >.< Soham Chanda · 3 years, 9 months ago

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@Ryan Soedjak instead of calculating \( 7^9999 \) you should better do * remainder(7^9999/1000) * at wolphram alpha Nupur Prasad · 3 years, 9 months ago

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@Nupur Prasad as you can probably tell, efficiency is not the point of my post. Ryan Soedjak · 3 years, 9 months ago

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@Ryan Soedjak Brilliant comment. My response: what do you get by spoiling your time like this? (as you can probably tell, neither supporting you or insulting you is the point of my post) Paramjit Singh · 3 years, 9 months ago

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@Ryan Soedjak How'd you get that? Tan Li Xuan · 3 years, 9 months ago

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@Tan Li Xuan wolframalpha.com Dennys L. Agostini Rocha                     · 3 years, 9 months ago

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@Dennys L. Agostini Rocha                     hello, I actually used mathematica

Sonnhard. Ryan Soedjak · 3 years, 9 months ago

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@Ryan Soedjak Cool ^^ Valian Fil Ahli · 3 years, 9 months ago

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Ryan S. seems to have made a joke out of this question, but I must confess there's been something bugging me about questions like this, and I wonder if Ryan's answer was really so much of a joke after all. To wit, is there is use for information like this? Is there ever a need to find the last n digits of some humongous number? Is it just play, or showing off that you can do it without a computer, or is there an actual application?

Put another way, in the real world, would we ever want to know the answer to this question? Would we ever not just do what Ryan S. did (i.e., ask a computer)?

(And please, no "a psychotic wizard kidnaps you and you have to answer without a computer or he divides by zero, destroying the universe" types of answers.) Christopher Johnson · 3 years, 9 months ago

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ans is 343..... asking how...??? well its called answering by 6th sense Suraj Sonule · 3 years, 9 months ago

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