How can I find out last three digits

What is the last three digits of 799997^{9999}

Note by Fazla Rabbi
6 years ago

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By Euler's Totient Theorem, 7ϕ(1000)=74001(mod1000)7^{\phi(1000)}=7^{400}\equiv 1\pmod{1000} Now, 79999=7100007=(7400)25717(mod1000)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=71431001=7\cdot\boxed{143}

Daniel Chiu - 6 years ago

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How do you find inverse of 7 modulo 1000?

Nupur Prasad - 6 years ago

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Find xx such that 7x1(mod1000)7x\equiv 1\pmod{1000}.

Daniel Chiu - 6 years ago

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

Sonnhard.

Ryan Soedjak - 6 years ago

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Wow, what an elegant solution!

Michael Tong - 6 years ago

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hello, I think so too

Sonnhard.

Ryan Soedjak - 6 years ago

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I agree

Aiman Rafeed - 6 years ago

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trolled :P

Jun Das - 6 years ago

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Your profile pic and message see to have an effect on your posts...

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actually it's my posts that affects my avatar.

Ryan Soedjak - 6 years ago

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Amazing solution. Nice thought process. :p

Dhruv Bhasin - 6 years ago

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Cool ^^

Valian Fil Ahli - 6 years ago

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How'd you get that?

Tan Li Xuan - 6 years ago

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wolframalpha.com

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

Sonnhard.

Ryan Soedjak - 6 years ago

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instead of calculating 79999 7^9999 you should better do * remainder(7^9999/1000) * at wolphram alpha

Nupur Prasad - 6 years ago

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as you can probably tell, efficiency is not the point of my post.

Ryan Soedjak - 6 years 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)

A Brilliant Member - 6 years ago

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Ugh,you took up 2 extra bytes from my cache >.<

Soham Chanda - 6 years 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 - 6 years ago

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How can I find last three digit in 56^789

Laxmidhar Panda - 1 year, 11 months ago

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ans is 343..... asking how...??? well its called answering by 6th sense

Suraj Sonule - 6 years ago

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