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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
4 years 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 $1001=7\cdot\boxed{143}$

- 4 years ago

How do you find inverse of 7 modulo 1000?

- 3 years, 12 months ago

Find $$x$$ such that $$7x\equiv 1\pmod{1000}$$.

- 3 years, 12 months ago

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

Sonnhard.

- 4 years ago

Wow, what an elegant solution!

- 4 years ago

hello, I think so too

Sonnhard.

- 3 years, 12 months ago

trolled :P

- 3 years, 12 months ago

I agree

- 4 years ago

Amazing solution. Nice thought process. :p

- 3 years, 12 months ago

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

- 3 years, 12 months ago

actually it's my posts that affects my avatar.

- 3 years, 12 months ago

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

- 3 years, 12 months ago

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

- 3 years, 12 months ago

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

- 3 years, 12 months ago

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)

- 3 years, 12 months ago

How'd you get that?

- 4 years ago

wolframalpha.com

hello, I actually used mathematica

Sonnhard.

- 3 years, 12 months ago

Cool ^^

- 4 years ago

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.)

- 3 years, 12 months ago

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

- 3 years, 12 months ago