Zeroes

amcavoy

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.

Tide

Can you determine whether the number is even or odd? :)

VietDao29

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,

Zurtex

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

ramsey2879

Your answer differs from that of Viet Dao, who rearranged the problem as (4^5)^6 + (6^5)^4. It seems to depend upon which interpetation is correct. Is it the same in every country? Also, I am not sure which is correct in the USA. I understand that multiplications and divisions are carried out from the left to the right. For instance 24/3*6 = 8*6 = 48. At first I thought it be the same for powers. But then I found that reiterated powers are evaluated from the right as in your post. See http://en.wikipedia.org/wiki/Order_of_operations . I am not sure whether this convention is international or not.

Zurtex

There is no convention; there is no right in the USA compared to anywhere else. You're just supposed to put parenthesis around the binary operators that are non-associative.

Quote the original post and look at the LaTeX, you will see my interpretation of it is correct.

ramsey2879

Then there is a problem with LaTeX since I tried writing [tex]4^{5^6}[/tex] three different ways: 4^5^6, {4^5}^6, and 4^{5^6}; each within the LaTeX coding operators of course. The first gives the same result as 4^56 and the latter two forms give the same result for either choice. How would you add the parenthesis to the printed form in LaTeX?

Zurtex

There isn't a problem with LaTeX, "4^5^6" is just really bad use of it. You could do either:

[tex]\left( 4^5 \right)^6[/tex]

Or:

[tex]4^{ \left( 5^6 \right) }[/tex]

master_coda

Of course there's a convention; exponentiation is right-associative. Implicitly, 2^3^4 should be interpreted as 2^(3^4). Of course, that still agrees with your interpretation.

Zurtex

Really? That's a convention? That's what I intuitively thought, maybe I just noticed it so many times and it came to me without thinking.

master_coda

Yes; it's even mentioned on the wikipedia page that ramsey2879 was referencing (and on the associativity page as well). It doesn't seem to be as well known as most other such conventions, probably because expressions like a^b^c don't occur all that often.

SteveRives

Zurtex, what software did you use to get this answer, because it seems right that the answer would end in a two, as per Viet Dao's reply? Yet, your software has it ending in five zeros! To help validate your answer, calculate the two parts before you add them, inspect the last digits on each, and see if a rounding error happens before or after the final additon.

shmoe

Viet Dao is answering a different question- he interpreted the stacked exponentiation in a way different from the standard.

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.

Tide

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.

master_coda

VietDao29 was interpreting the numbers differently than Zurtex; so they got different results.

(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.

Zurtex

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.