How can I find out last three digits

What is the last three digits of $7^{9999}$

#NumberTheory #MathProblem #Math

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

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 - 7 years, 7 months ago

How do you find inverse of 7 modulo 1000?

Nupur Prasad - 7 years, 7 months ago

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

Daniel Chiu - 7 years, 7 months ago

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

Sonnhard.

Ryan Soedjak - 7 years, 7 months ago

Wow, what an elegant solution!

Michael Tong - 7 years, 7 months ago

hello, I think so too

Sonnhard.

Ryan Soedjak - 7 years, 7 months ago

I agree

Aiman Rafeed - 7 years, 7 months ago

trolled :P

Jun Das - 7 years, 7 months ago

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

A Former Brilliant Member - 7 years, 7 months ago

actually it's my posts that affects my avatar.

Ryan Soedjak - 7 years, 7 months ago

Amazing solution. Nice thought process. :p

Dhruv Bhasin - 7 years, 7 months ago

Cool ^^

Valian Fil Ahli - 7 years, 7 months ago

How'd you get that?

Tan Li Xuan - 7 years, 7 months ago

wolframalpha.com

Dennys L. Agostini Rocha - 7 years, 7 months ago

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

Sonnhard.

Ryan Soedjak - 7 years, 7 months ago

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

Nupur Prasad - 7 years, 7 months ago

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

Ryan Soedjak - 7 years, 7 months ago

@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 - 7 years, 7 months ago

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

Soham Chanda - 7 years, 7 months 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.)

Christopher Johnson - 7 years, 7 months ago

How can I find last three digit in 56^789

Laxmidhar Panda - 3 years, 6 months ago

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

Suraj Sonule - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$