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Zeroes |
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| Jul27-05, 10:33 AM | #1 |
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Zeroes
I found the following problem online, and can't seem to start it.
How many zeroes are at the end of [tex]4^{5^6}+6^{5^4}[/tex]? I know how to find zeros at the end of a factorial, but I can't do it with powers. Any suggestions? Thanks. |
| Aug11-05, 04:15 AM | #2 |
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Can you determine whether the number is even or odd? :)
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| Aug11-05, 05:18 AM | #3 |
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Here's my way of solving this problem. You notice:
[itex]4 ^ 1 = 4[/itex] (end in 4). [itex]4 ^ 2 = 16[/itex] (end in 6). [itex]4 ^ 3 = 64[/itex] (again end in 4). So: [itex]4 ^ {\mbox{odd number}} = \mbox{end in 4}[/itex]. [itex]4 ^ {\mbox{even number}} = \mbox{end in 6}[/itex]. And: [itex]6 ^ 1 = 6[/itex] [itex]6 ^ 2 = 36[/itex] So [itex]6 ^ n = \mbox{end in 6}[/itex] [tex]4 ^ {5 ^ 6} = 4 ^ {30}[/tex] ends in 6. [tex]6 ^ {5 ^ 4} = 6 ^ {20}[/tex] ends in 6. So the sum of the two numbers will end in 2 (6 + 6 = 12). Therefore no zero is at the end of the sum. Viet Dao, |
| Aug11-05, 12:53 PM | #4 |
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Zeroes
I'm afraid I don't know how to do this other than working it out mod 10n and finding the largest value of n that is 0 and all previous n are 0. Which is fairly easy because of the form you've given.
However you could just calculate it: 4^(5^6) + 6^(5^4) = 15394461414126262391327387935172665487700414647804095367550129093166641 6344673\ 16292204899771887865625138809699534877025687801079151971613770951000764 3030990\ 83258807703965889412203972832847386635903683923597028003963487946206680 1632572\ 72942092410165365541472378028092109127807479675525833585950478206465524 6931712\ 91845646158605028687006946849557148169516364312530852786806028663589488 5529429\ 97418202886152600868546536970917571757493233253847138531517393891509187 2235024\ 41116742292971017605949593399447912588076743136693590844270808079594597 2525035\ 74143255422990106699667651334806559096070793349857371848825755042986390 4837918\ 16704672660738056850514506400376770024086112713293022294200509131494489 1939965\ 91307798405375322196247803965459304307585736156739402510416510500423927 0622408\ 42515524652072791612022000326042237224814981789624548527521525285185824 6200426\ 57579135554250432982728417982790117692632513000938472223938495412225707 2123977\ 36928950988701878141872105484499402816114293256198548354265707541789761 3128253\ 73209992798497695415377317331169635703377086486002085932083634271146041 3208279\ 92215134632680067965457699607980761865814038422067176258858997464083635 4391118\ 99927096626714069306827835987797205357513332625247118220567717226539593 8530805\ 23770133444591903198111736784581056889808626111722577197258923206857456 3621699\ 75212140024121106750964357403818571238045303222671646435077964506411326 1118483\ 47104673914307283261207444064426090677332242319777122026854284691940840 0573268\ 68091338935022779833431847717994219704146523625681123276670943714782420 7886347\ 38437349401998322952378075023035848327730427768719953361495782384309014 3034353\ 13715804816419189438760112140493111260599203130659677763123101412392644 7242581\ 80374911074980182152877712439853943458761484132553665053536028947433711 1310725\ 98077102368878647823639025506358087942587663959631579552477165449762042 5140613\ 21215105715210847728279557726359672657577247966622509178121665826756744 2716753\ 97716459396343478362946770052024326686674885006864419384399960022115117 1679885\ 82369681108089147459583237164401662142256373274182256127694299983814049 4949895\ 42287376180416288032618279583631384354639665619723286374299106321494200 1526287\ 77670661189967595252581794387819415034108236270956274631836811609137436 3759731\ 27350496498508581234042639880339453551170973643842207015865975273416044 1021332\ 74392603663732702764129106365261805621245211384878015552350019977714983 8211343\ 41417893722671993275062599460321706731629830010880331076983490397329198 1793957\ 54017792382761604394085047107877392098821295017311920010669628272082671 3693134\ 89776499567797716875131975919189873179199138016136396500404853591494487 7448406\ 27001619933926605675992570468697317825201944453103330050868047596226017 5319630\ 57421688945042548813174361117001088184932670537968703005827833435894304 1296971\ 93127663148845793863267602594309631635014838495310143451781975433710257 6396497\ 10667195530090280677163021144543033853733162160979488004745098986274591 0086669\ 88197366981683338290883722169295799717640054841047431460209804719115155 4568553\ 42248749503781630015068260773443082915053443743098581316578517271150614 3640188\ 13511659886381484417192567072074510377009647735766058866336206492456827 3659588\ 29694687340205985201633219021479507923557336289609305795966618756222720 2448342\ 86679325161452214807282600447981331158966598315451153237270789522090396 0250376\ 85386936728993436732490915295281734211245770414328728638794424269198114 8289170\ 46987700506459251151757961451041422870365315131035974513781285347633163 8040601\ 48000957245735416385696481922136495241347325400854274344095236919992153 2070721\ 14537666196390559071768227440198729193202633273038303542158232641389782 5455637\ 77038634931594548938982322175843674824315141765176365696441987448851017 1861332\ 09772387816749750969239360174551441299659001171577382388577942134190380 3444823\ 08822239742272481810039371791506111510158581191090457218807437899049753 0294138\ 07040536134163747638074391327169617780252478255205178367944873444903572 5575818\ 17467447417847119086558941321083861830209369514043150440348171204778760 7752575\ 46048656786974434812889885327742445270624721889655681976109228737885620 4305287\ 79811410102670897947845330363650041494183510932496895588275726812701255 0191149\ 98136271696588945701955379864500846512724315891005088350763659061876087 1646980\ 18543394191744374850565777121736340930485155216640032478295779787920261 2743409\ 98179290660116991408100181189437161663177901046330323266960166199640982 4659462\ 97073112477196775846837519979207208321262807663122779602511961153442490 3265409\ 83443231793211028217201996264866472608526954029130605013239036789816111 1653926\ 22167249212623398690683772259429412865329761114309784946879653014881558 5895774\ 20862054787423691912496193559866418171490587842576657083754908713910098 7030922\ 88666920407298556783531277270848360068816301235457805470323715288090531 8631705\ 74301098357653968664688721301362709032121230794153353200105453365121964 2233160\ 10478399455004478657080381793966898627712847767950086661160413662208201 5227895\ 58251878280252318617023065459254806947864706680703101099230970503225341 7605914\ 55771852223343571543499980046722351496677909343402059951191097697151415 7605670\ 94688864169717235233489667517755528009239554268654430133877165947301940 8154627\ 62557896570928639005221391720506420806946415866169474389116948289688789 2038633\ 80018476977282328146753976005050991522276104724698772556538257221659665 1272198\ 31411849451240962794194494373093096784093421600484607117140527153852666 4813271\ 90700137492952385295146827785961500607013731479440275588063147418139233 9629855\ 62087627942827852936118592792110672670978126530180860950939028714550535 5281932\ 03674987133708807272092095551854513089983342412256937353808929617181208 5128671\ 96108224463379118002828578729164281452360884755762377953436460077836322 7950448\ 02243920147922769888759573246633216354729103060362509673635504365305951 6636009\ 20152080888622686819024761963281733688353875787041410011888634930303222 1097792\ 16942836928446925426829822802326691253562172865994137736186664502141365 9574035\ 14161873720087637423102462989707868817234599976528375113518608513595138 1652773\ 99722925982140264300541760271072863806237781640010130549830263199170433 2892515\ 27336297543938658532176134243423582682551206770587101500096904221725780 2789601\ 30995990666274588358786344704141625253812485579880257688593252676072749 3957919\ 57408349390710453078810136503984263856182231947221709795444687286558778 2312224\ 73115436010340493786586482189866981707247666134492464758093230639008850 9421804\ 88949802440692567492052521924688214972860516543820226111556209116252037 3087913\ 65646525572590309766569128028732213206817274894770829933737379781141047 7262233\ 77653403082035005582616067864923293379508925336509772888005977061235584 8661120\ 64488249678558371537594470712397261476081314945780654965285157213721082 8435133\ 04251989930664276716195853277418178511812913006068828996491287635725539 2409390\ 54194861104476362166093547185762037994482564583867870776764901279192749 4841722\ 64041850967814280639137018575115324715507649792552211032172650654504221 6017357\ 43489862894365749201915974417615568046100577764473339655954549508848921 5021163\ 89563038965997365099366281518449124675862835341654377339407643740505174 5929030\ 10410894095071906990324967491586381990395406145670627084466745644228182 9934580\ 21095707865104936507036648457613939829851593754927755250812879956204237 5208675\ 24306698994516361054246437182736272882554416730657930446298832808535967 1706863\ 93884474518482049348363767675782917511716165250083827491667039959291150 3502443\ 63720773470517503478195983229815193334460236766393780565989753865840004 7206259\ 46969754413489188428103798696494557702552371763313266451253668501350988 8688351\ 14604055996020734623800790262500988264323489909748482979861507104719335 6470609\ 67361905315780528298336903233114992380304496872557684231223090177088915 2307323\ 15627036887850988113956697016550678844503908643743874908326996711018480 5767783\ 57064479570346555852722626779983372783480218726495191969332872411774996 3987211\ 94677707820394469187032734278168532236645860036286774699565971581266168 6151699\ 33350665720132173866215559195614897223673943096550551595956713389241473 8035355\ 99438986088004332927744135081969228109406486836363348693698247338324425 0889524\ 37010653579642479478501234377553104674591021097449020448904349755961212 0795111\ 03388012429555705042088525550506070517386250583872148187629789486804752 5350951\ 42771976870652892884784278661275640848827603365097529026655010704371772 0598580\ 73156409487755402921574377863821013996937569949543589732361181679369220 0881296\ 11904298128811460437732186483430326685516533986945308980273302035082892 8619396\ 09771778265783916081959362235743971633266709148064319630224251974765770 7965201\ 11847629726059386353499520514280514310298756828171210424146835279052678 5789456\ 96279811554602718863250932051814634291482694590453218499567482731547111 2548808\ 87856276843277242445423075304616798495790006744235757783800967914007697 6845743\ 18510322264932984229271093037517648689448980331850928483594105387165193 1099567\ 16094643362224278806338304786189160828302136338278746919860471297900933 5135876\ 43492108044602677005631809711417070223483010190096023761673041938660956 2929954\ 94230562459501219075274264371423659910373599490839774190264879035087934 8228446\ 20273813252594235226793798464981162911629748444430914151662500270221164 2375924\ 42555489526290377065813107972578675798812460568540050188440706636047047 2196896\ 939250351868656941828080273920275853921484800000 |
| Aug12-05, 03:42 PM | #5 |
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| Aug12-05, 05:27 PM | #6 |
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Quote the original post and look at the LaTeX, you will see my interpretation of it is correct. |
| Aug13-05, 06:17 AM | #7 |
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| Aug13-05, 07:08 AM | #8 |
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[tex]\left( 4^5 \right)^6[/tex] Or: [tex]4^{ \left( 5^6 \right) }[/tex] |
| Aug13-05, 11:11 AM | #9 |
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| Aug13-05, 01:00 PM | #10 |
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| Aug13-05, 11:16 PM | #11 |
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| Aug14-05, 12:06 AM | #12 |
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| Aug14-05, 12:36 AM | #13 |
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You can check by hand that you get 5 zeros at the end. There's clearly enough 2's to make it divisible by 10^5, as it's actually divisible by 2^(5^4), so it suffices to consider the equation mod powers of 5. You can simplify first by factoring out 2^(5^4), then see what you get mod 5, then mod 5^2, etc. until you don't get 0. |
| Aug14-05, 12:38 AM | #14 |
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Steve,
Zurtex's answer is correct (I use muPAD). Viet Dao used a different interpretation of the exponents which I don't think the original poster intended. |
| Aug14-05, 12:43 AM | #15 |
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(4^5)^6 + (6^5)^4 => no zeros 4^(5^6) + 6^(5^4) => five zeros Someone else already did point out that the answers are different depending on your interpretation of a^b^c. For what it's worth, I Zurtex's interpretation is the correct one, unless the original poster did not properly post the expression. |
| Aug14-05, 04:40 AM | #16 |
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I used Mathematica to calculate the number.
But to just answer the question, you could probably do that by hand as long as you could work out 5^6 and 5^4 in Binary form. |
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