Statistics

Problem Statement for "WickedTeacher"

Problem Statement

A wicked teacher is trying to fail one of his students. He gives him the following problem:

You are given a String[] numbers containing a set of distinct integers. You can concatenate some permutation of these integers to obtain one large integer. For example, the permutation {5221, 40, 1, 58, 9} can be concatenated to form 5221401589. Find a permutation of the given numbers such that its concatenation is divisible by the integer K.

The student doesn't have a clue how to solve this problem, so he just answers with some random permutation. There may be multiple correct answers, and maybe the student has chosen one of them by chance. Return the probability that the student has chosen one of the correct answers and return it as a String in the form of an irreducible fraction "p/q" (quotes for clarity), where p is the fraction's numerator and q is its denominator. Assume that each permutation has the same probability of being chosen.

Definition

Class:
WickedTeacher
Method:
cheatProbability
Parameters:
String[], int
Returns:
String
Method signature:
String cheatProbability(String[] numbers, int K)
(be sure your method is public)

Constraints

  • numbers will contain between 1 and 15 elements, inclusive.
  • Each element of numbers will contain between 1 and 50 characters, inclusive.
  • Each element of numbers will contain digits ('0'-'9') only.
  • Each element of numbers will begin with a non-zero digit.
  • Elements of numbers will be distinct.
  • K will be between 1 and 100, inclusive.

Examples

  1. {"3","2","1"}

    2

    Returns: "1/3"

    If the student chooses a permutation where the number 2 is the last element, the concatenation will be divisible by 2.

  2. {"10","100","1000","10000","100000"}

    10

    Returns: "1/1"

    No matter which permutation the student chooses, its concatenation will be divisible by 10.

  3. {"11","101","1001","10001","100001"}

    10

    Returns: "0/1"

    It's possible that the teacher is cheating, and no correct answer is possible.

  4. {"13","10129414190271203","102","102666818896","1216","1217","1218","101278001","1000021412678412681"}

    21

    Returns: "5183/36288"

  5. {"139012963127893461274126478482"}

    1

    Returns: "1/1"

  6. {"139012963127893461274126478482"}

    2

    Returns: "1/1"

  7. {"139012963127893461274126478482"}

    3

    Returns: "0/1"

  8. {"1390129631278934612741123112125552523126478482"}

    18

    Returns: "1/1"

  9. {"1","3","5","7","9"}

    5

    Returns: "1/5"

  10. {"1","3","5","7","9"}

    7

    Returns: "1/6"

  11. {"28471291","3125215","4366435","7125","66"}

    13

    Returns: "1/6"

  12. {"1428471291","3125215","4366435","7125","6666"}

    6

    Returns: "0/1"

  13. {"4366435","7125"}

    5

    Returns: "1/1"

  14. {"15","14","34","16","171"}

    7

    Returns: "1/6"

  15. {"23944","617211959","4723869","282208","22555","63","967","324","7","4796892"}

    88

    Returns: "16/1575"

  16. {"357","541945","920187","35553","554","5304087"}

    49

    Returns: "7/360"

  17. {"153579286","66283","7885492","9884","806","58136246"}

    69

    Returns: "7/144"

  18. {"235306","148981333","47855855","502419","4938026","23563035","49755","191","398"}

    49

    Returns: "1457/72576"

  19. {"720"}

    40

    Returns: "1/1"

  20. {"4328548","60497","696064","3631920","3366019475","379","840691856","3643813"}

    47

    Returns: "167/8064"

  21. {"3829","9548045394","75853","4377970706"}

    26

    Returns: "1/12"

  22. {"6","7630811950","3282953866","8133","27","425427742","402965588","1183793"}

    50

    Returns: "1/8"

  23. {"8723998126","4","855","73","7926164900","831789","29990","39","804240494"}

    92

    Returns: "551/72576"

  24. {"3","4","40","4259","173249954","4120399812"}

    76

    Returns: "7/180"

  25. {"1","86704623","860430361","62173","912882"}

    47

    Returns: "1/20"

  26. {"1","94","1194960","906","135","3432888","42301","62214"}

    86

    Returns: "101/6720"

  27. {"26","651","892345194","91","924076253","547625","475"}

    19

    Returns: "239/5040"

  28. {"401529","92","359","85","10884","29116","852743296","8242074","9213176","3"}

    85

    Returns: "1073/181440"

  29. {"492287","205","30","9982295402","83724481","8"}

    35

    Returns: "1/20"

  30. {"530884951","1992","70464","75230","8579","758041339","15"}

    33

    Returns: "2/15"

  31. {"984","396","48385","6","549468","4582"}

    19

    Returns: "43/720"

  32. {"1350237423","9732612","57866700","8642096235","7682200907","6","20"}

    17

    Returns: "1/18"

  33. {"3","70","6","9900055822","940563","26101703","77868319"}

    47

    Returns: "103/5040"

  34. {"99","9054744070","551","213387028","53230","820751616","686259477","7372"}

    45

    Returns: "1/4"

  35. {"830","69","75","762006257","829177","5","412751918","169933","9435847694","5800","95","1","49948","6020432"}

    41

    Returns: "9483037/389188800"

  36. {"7563854","55","1","15577004","9691590920","569","36676347","9624196","2852993","95852523","9870565","6441622426"}

    69

    Returns: "1487521/34214400"

  37. {"709139764","1","17423","84835","81","7086721384","92048642","42"}

    8

    Returns: "1/8"

  38. {"275537797","42","30994","551559","69369410","243011","6","70","5"}

    91

    Returns: "53/5184"

  39. {"104","32234","561","596","2620204","71488181","66652","724","176228817","6316","576","15198967","19284","563439188"}

    67

    Returns: "650577247/43589145600"

  40. {"30695","3656","35126","2374748","4372024965","77","53","871340301","45134","52","4","11","948246","397864","621"}

    71

    Returns: "6139307021/435891456000"

  41. {"31105478","262","975056","97525","9396311397"}

    91

    Returns: "1/120"

  42. {"9642063","9349126","2","95156","50931851","32418","3456989","14204","154861","77014638","606515952"}

    59

    Returns: "4181/246400"

  43. {"4955","951","169753763","241","6639043","4975036","308653821"}

    52

    Returns: "11/1008"

  44. {"3623555","75824065","5","9993173149","537248","98219144","6838626158","881849","610418","830228587","1646157","9002865","744"}

    38

    Returns: "15757277/778377600"

  45. {"181174833","9818015","22078586658314","34992439","49822","8254715466","1434642685","4910101","2472512804","25","2231146327","91604","484","54274182","95189465"}

    91

    Returns: "898596781/81729648000"

  46. {"407009104","266552","3041","1670028"}

    2

    Returns: "3/4"

  47. {"51006","10453947","3","89815063","97983","46","777","62657","9402","6925730221","28421","67","295964","20765","5056"}

    86

    Returns: "844734139/108972864000"

  48. {"34","22478018","665454617","246","302476803","5951201435","6","44536111"}

    5

    Returns: "1/8"

  49. {"394","78241","553","755234","543284","83"}

    3

    Returns: "1/1"

  50. {"4","82153019","89718","294","726033","3","423","851651","712789","9634835440","269670","937087146"}

    64

    Returns: "1/120"

  51. {"55878872","4541771","3362536","635501","13","44","290416","14","902807","226426921","221810"}

    2

    Returns: "6/11"

  52. {"85","5","340","30","703812033","242","501"}

    7

    Returns: "31/210"

  53. {"28","3","8132993","55752630","85","537360","740","2","58","5171837","30272"}

    79

    Returns: "505093/39916800"

  54. {"195","634531488"}

    11

    Returns: "1/1"

  55. {"9024117813595757685542600743751348876347330432567","1197389737464779456073623171046082573029814737","78544148475854468430994543175744855737376228339","272800327876775815470812055184812594083070747598","5894476784005780030612636667906453662266941775359","747350905643261798666584543400475849582707960","2896353837594915669291663667003937638710","8688418431498102702229298075639431273900929118","176178519099031799732145093701754584291915","2309408070004018896242748884829056954545222264","32078582673571390543283110291647577796861859","9742565105291537883407784178415600848372020044"}

    8

    Returns: "1/6"

  56. {"2344897230915641676531274003837426074115845891","639362183517537771625268080723478402200422612018","63707211861855396462999570030845536407445095397","8832974287132282141338505442652895921402138","963420496313851717600737122333915476091527832003","531223667389694322824045304758212216280713251","51757303573969335925648731162399165581969912","4172960281752121944598946252853927781460428805718","4384657537872980311428454716893415381469017460296","25996505058731498186288911259648649585792136249","63363849774150869916053947944823805015454928256"}

    7

    Returns: "43177/302400"

  57. {"38472330652974487345630375181345113034548811042","1364008938898028233405466335442348230435912955","859598371850182836397132439210183402886824","3000076855445699330494941253616231121785869619","392527404237248879576677096267620295459603306","318149714750839561782861050252017955168117273345","515915953614461329328232830028675171472676416170","1422994255075919839236287177867429139943","8304061029805525786010893488142477580430100301652","86158063145360568365893814561213253672592391606","91533553942276653857372012286868163518787530","48914897489101425953667281966992042627469111749","72865020427986330904864036279024756419628048550","6343036432689386643400259591101143045002933237399","8929790798138498419249241508053128589557770"}

    8

    Returns: "1/15"

  58. {"712985740303991962092422447206770351792644220","473366184444449057389292426289159907179885670306","2414186064146847577083189612551209171636","745444217931560383521701918115819543397297314837","610479415834571479037737228304265813584037553","1974208433963669542240430670738784591171037710","987479485653601016336985276681657661243244268","9641759274315860205930904942941055213211706051985","4822637224169217321772026147216319736140906","6328863355416562575072353394629048974242469","283637745042811504900506128177984519952154861969","5277807804500208281174260359289139477918183214"}

    2

    Returns: "7/12"

  59. {"303861612098251554849682845192084124914874240725","9287663384999770204930177889924111388969864","7096819068123999896387916344969420062836277729840","1952660721045190427425548212230302252798388191","504868820338004155888299862670555017062403972","36465500494948570755943902530480552613635","865015054621872172855769330701992548550413920721","9455951380213021204500790744137584986290211747477","73846936175052578935757237723130858129995","7112769975543901888533232114829222411851732066541","4545977811535973737608175647797479808277973527240","452543837971349289024474665371952981193322"}

    6

    Returns: "5/12"

  60. {"393735375036109534283588511315661231708128","57943842466324080756537179545760169023745","822086380662228572691187238051837786633537844","426099545989515149608620827062529466428638","91940092037062872149132910519064915114813167407","737533626039614037333703554574769370844930","9777075827146775303360922445097890889538350830","8312356788914382167178456280658531022313","2128246541308044853880470548303465431395","91543534561273525372149248152269119163104830945","9733376630456778942553139827023315235355401038","82345582334182917137497246485306406794560","357794318851960393967661242291917505908416844","583085466684340855200078612070997729386231936175","622053920959780288953120105429779206436510931225"}

    5

    Returns: "8/15"

  61. {"90889110132225498910296419988559742756286400","227245467950672711084940963017765749831709090","1505360687626649489234052924655744688619","4562787558955681813230899329493214074613135","58953977613914740237900871667505610644777","410961360253311927555829534892711922789604","6111663368972604544856410227491344743616875660890","461578913025190262053946857053377533281590213","23841781986304418776730771661452220041414132110","3662250269443873146969829179177291642615225183","4085495607397363958316762795317666020593","4629685447102843995506412352080417650769686050"}

    10

    Returns: "5/12"

  62. {"889685223792008158839102388123182167242988590484","9585194839102543009705724853931870981679722759090","3909383203767197323517077809013274462314","4996035150174600631412470095618870264864","6206775039071259659497469460727335686674","2703430784548672816903959695844168528233366","9443794022237090226838708129963183464483962","2134255265569866812231431894656325730983997","315273957714882026076511718492734765560405036099","2759452295404719883906871681802775640668419810","1734173449185794406897634295386252554332526195"}

    10

    Returns: "2/11"

  63. {"2326546459631830379044037152374221211241547862","28901169649352585391207955587521044895305","58388234735498597756639937132890514387410127039","7257937848944609271390491778664365835710691496","8768246250198506032858439693020727702478011","1187101677469962328562492418446353594986221033867","37078122192462039953402915192901993623875862","9626496308266234935821220555256407136871","314879433361282595889921792636949138409750188","391943863140558274141772801903736482829583","50846258644416942181539386191063448271195314923","1277440298861791415925825270402938847185941588719","2588133093573703881034624554460260707963"}

    8

    Returns: "1/13"

  64. {"97057769010012559755465224875066025436411","66415991080962526843754154078528733731299","77356035335419204015992982061582167503815","86968922849824037288955997359855136521618584","87064771981568686528295121243321086008179450637","206135068853091639910616783796373700148065","8009200119159412875874300033833200871903142348","8098226804181838666382504116159126723234203777977","92405804188943503954441466699661153011365978","95436425094785471779393585210520596970752801672","3194131623972977124875736888853062027420807"}

    7

    Returns: "4043/28350"

  65. {"58638997982496333278746635342536241001568655","93578039392840572277257107896056878279110","511443810991112326269055826930322523019483185124","442731345380718673146387569322607655443864845488","215245369356727758880814495110279991912828137153","99788612496537708841016642584819056294555981","6626784363451377778416504294406903567371097311115","26641899160148388647862298627968998862448326","22868579200955579936684466904245796109377","502210030724833176682181305305179152554720","1301350408264523575752160947172134381827529","71661433830048385767986762588913740082157","59804963237968119568098929189730828751264630737","9228423696606106764842123952163193114144"}

    97

    Returns: "898746451/87178291200"

  66. {"307904871138505730318515215512948113037657","2156545284127759596223687957674052689275504333","7775759132431354411447868041177259606425183","4712390349238793697754143388112998322422532606","316997234465585810185329812078459692858116","7448097408606880767209051558649271086195","8422904215641732449767808687989247436327066537167","857008154597247394433695885983957103277205112395","3192765230584796760560255014395567457694424","2071499781597264429363666280120968003919742","3685009180804431827259394133260967512217110","64576339833599284417098544158610965816730337","45767142092521322382082046387663387526250958","7465478643726735913747030728465469633427575212502","648218943007128084826629782600353457421847230"}

    71

    Returns: "18418031623/1307674368000"

  67. {"1050668282001487027293593188927657532090","9253632649521281318550206955438356681782499498","7233565068222011116246626931090292987276286382","530694034093510819668373346840767783471850","1564246893756179147364514599368809305013803","7711898799156398003470528349601981812532831804078","2741865026599996614429654982707232703476750","1654763727428810462966209016122094892990932293","1114425980875887038015282923165892841714","78216234343103745594437546271515891685860","61594654890077444242593894913228934258701327","158929146171242156457088616807662155608452077","51819060191879618684456464339121889741848","62881759331844298297680200561445785444857"}

    61

    Returns: "714582229/43589145600"

  68. {"3094181232450146485258822135075588407149227938595","5622778445136046339313804663826437479223","91671623463231744810849116526675455720633189","7908272312131692232221405895895639443013389","33412594489974301889912356925223133455336317","514895935404356890437945295183244962674009698","5457021906069860314351468794491298658010786","2207362784924285566836027059520088678520","52293944921514677595529737197791207167349273","553193284930206584340615857738368574554433169711","44089571085241180773915073468350605388015162405","183373552138915283739100179724255329019807635","76657928779610493510829268910493304418060363990","6231023799009606951304504556210343596813580864","21778353254567784543925429457176104750005969923123"}

    85

    Returns: "5128215163/261534873600"

  69. {"5593197435308260220493073396313504232480889","240318017941055431005882977960712173055485","6789851671157563800995467753656099982941297500364","8799240116153879245993429385280622554679115592933","806172227449094486769415258215937005216869514","8874783789268381254207630886788912424155","3257953962574633982077168109663679079582880707008","953830354331717915953412003491995683725323","19317518467204643659519328451891764475534467226","99594791794317774664474851245791832666228","5920445378264089042197898839963812382091","926251280743580811463613053416535091043378","99243428425522519949759874983893915233963458140","36835526289839623366527829865345549076077325","2496410048256100776277186816708433364370419716919"}

    77

    Returns: "27614131/2223936000"

  70. {"3930840882424407767842049930433423467436534","3187868194444379072088458493760177460908","4059594787657333744021416723359575291970427","58132098055803617824858107939371023385103299","26188062479114830086485669101031648811371166914","71385979849099241058596329711779761061456","97125881294637065489504247019099975822128","3247324214251638867878450289493073872810135","1361238299690856512675375399104741498684","787672495178281458822997328147911158373422454"}

    52

    Returns: "7159/226800"

  71. {"37204223658112003704435350335577619232085952908","5112865444501021762446585109117634160188","11111605461205167010172341314881490759705152370","3385579105393354734521314914411347412736748441901","121758145760119239873323663904016156708591","5033203276110350581164325261932368530840","2253919660258149831195056500395017886256","8177971396636939322169122683498814682789348709","640204656575027598622849043039908990951340","46677205285523272308727289091129937041079","20522492570688866497667050942426482557665105048","91976902655929917213315804248405603259107","5388807571486465206962890005398894497049460246","55582803033148246594948368572852944128856"}

    74

    Returns: "58231/3057600"

  72. {"65033124195940727045000162628436023174006897","794985386923275262086630198905652487658179000","479689442046965460800999323963165890254633171222","983820045292547031408286210614967644543326302416","14697255351838542124890375195624795894949","3898328326331307094689473725205540345403383860","943496370986565231600633003206565558841281229027","10049506735675439995927703705421358353339482423","9585651076406301182977341035253870131354308082","4098363479144753942093953058486174902121728691","8095660829781824872671936794978547523354","833661133129912896374046334037401425232459","500433991016372119309373167157831041739332370","2897787362740286457868965548920355635218920","33677101589924219927867874016858685404716848554"}

    82

    Returns: "798896617/54486432000"

  73. {"706204256789010440883989714886081720590740","141528466459436173773245890292575311683529","94787839835973420825698793083892516225221858030","8441590942960482721105129513194701722784","2348297184720431029405957597769716955606796","201590265925922699325947247433039456011981","8508223712319226085353216938940545662810","756348129702085143166219870423363187171419","870257329624644166718702667348933906765793516","69560440747040291615047989433978475558376625","73080238483396671454888572320030472815912640","785152154614395107240442542351690399065284","3905430130108387085511258148198925989927669","5716838435089999287208418193122885589601616"}

    89

    Returns: "97957313/8717829120"

  74. {"43748946283200409729282100025619621736634007618","714813770144242463502397492284782920936021690","468100466425129555457066638331988288204986","961312313212709750719628511041686115590976","746576450560846584866385700506958142664188282","823894995706059408295965154281998859200285879","7407533523775945749033959957597407260992","1783797552294348647541308331189974071237983454028","2744122676476404893707053342190472556011","4869685249990219904741846433398496774751025877572"}

    52

    Returns: "7079/226800"

  75. {"85273893967597002669432307452802017566696654502","32063856761008922197794873119983659705871753","8786000494615110279448653688864129871328731144","241909718777476639405051826504320607111573453","11734509037977026411304534647928257282440567","605001618101051335516727888534409615651439033478","5475392232545289889191789045530435623582173","692133073597875322115913829519348116334448","1450122854880963063006341135598506763370","441540923378651820827409424096584957854986237129","6786727533482197749526916018390833206597059052","279413000505634443608240351292797136454195580","25119638044568498585914068979266709624389185","52698769602622807779246059796949179060909144077","9210583360142684359717677476968052974194483811042"}

    80

    Returns: "0/1"

  76. {"7014787422435059552134536144375653037540394984","10209985321910611967144981338446481628310","42910852683101993123793627074431636365","8259144953704001089243688554003","28618095566348581688806499","548638997982496333278746635342536241001","68655193578039392840572277257107896056","78279110841144381099111232626905582693032252301","4831851248442731345380718673146387569322607655443","6484548881152453693567277588808144951102799919","382813715349978861249653770","841016642584819056294555981966267843634513777","841650429440690356737109731111541664189916","2483261128685792009555799366844669042457","7109377250221003072483317668218130530517915255472"}

    98

    Returns: "10674904843/1307674368000"

  77. {"773367710158992421992786787401685868540471684","5540175228446816506620442690821753500337886728","4446443051271920433956446674387","526507211072075719755684375888095","6003750363780052875345190197","3085650231429589965339974880314503","70200207016122005304607733648457176059035009142","594923237713852283337999576","2149167264385889622985761638839","944407446674698600851461675694239315845940142","71024817281510594405811208892717314816107452522","117887732715088559553055178159627280619500","9636794473918395920342693563018163495","8933227634940432","229237624486310025985149849571281664886865"}

    100

    Returns: "1/15"

  78. {"9617810542343330699361771866806887486307","11160728658361161583120531671527","233491355476098701575583160600","955084666052288697614218963994169211342200","6492531148798852204979163238075126771000","600543662668464244468207734575895313","9983825389945180795635139191370","1672548214673272235169099316317107341","2519692388301654649488547817","46449906115025431825880405475853454319436900","1815865692304273280998705524313912434903","7291455940076612539615431315327161076897590","2135294610400336207198698278835539654268806803","68105075450644921332340160402","207076610475056490319938760621104066112004"}

    100

    Returns: "4/15"

  79. {"2308774779946980720300316064521","5467565684563932777457543860324306261205498","43675597668651596933435200","6511853252446282945534710846798730619959608615","6801654930810782718290865957153390","23698048015987410689476053946421154159887","126933945430042379689819621250552555867606381","95222661975904619187982657336","65646331935802642326522146919731280991795214536","8069606527271974010135383738942488089294329947086","751900443875080297583668531","747571415591801812766340538026282","5259153740415976537118684802190234151","14206092503301067398328330890","3080493457289088016159081104701573887583989319268"}

    13

    Returns: "12572878337/163459296000"

  80. {"49860318618333488435448981383056","3675512279381861725374946591123892374833643","41917432129966508789820899146135","60624885380431063596888513184799114459533650","986424604128906023501540477363873","429122209488484527726973456527064","498031902992598606425712853460128680214461624943","114031806434695776244903724794650783928","276952721411727823867512146890","5042383773757146742811109123387207193323","65496588159066048816553350839292281391","1741229449354963457934551056491578533120073193","74791569305227919584859990143677588875447655","264263350923804706839367854650017316409487","9732225655468147110122327606750869928009"}

    24

    Returns: "1/5"

  81. {"695484828537883128141970049858","153220399933367642965845392148420290","7142183470277020158245284531717586884","226985866717720940999254681015619","1494489550191285120174015843326306974117","534569906152857047116655160080953003206294","5248971523192174187668998866367804121","8008597507526704175503581498956426135","1907950307427983208927237653666351","89735653034064474602580327116336810279","201002851213600783235514989156017","776767917099252779880756021103289283464","8829309954512421357962442103449","8446784897100273009344058539754759","3386453960323808452206822261193611293740281"}

    33

    Returns: "196/2145"

  82. {"3864624640196542872058684108","27559949475672330922366029973413","8464482766724240169547053005960","256241395422591567377679","36963053059246030556740033408372666","923783790744291721922092356989","5800883932738872716037727620901","267306504975366749135767","96016345739706436194361","516651574013925257757236281329285","6771125407419978629302704386008","990564041633148886435625443765","67415498374971794905442196786078854973","986045237210472381949479483494076703954894574","1069473244079792231274764"}

    93

    Returns: "10545947473/326918592000"

  83. {"46015225608091868981819810227136255","666905197238113866560481730412606","88739592073665923407197371415","7169507834093501055990111480150","607590001969781279611330840740","82005141745582077680989","227863686622951881782675365552733","1114061643664597059571109812","62872067284638150649932057","908567441673171781339535764275","740959518514817613627047744128126","1379863996269206938699129470558498","211389013999607938760443517","2497685331590271351188154472947814","394748847655699860825914252568648306"}

    9

    Returns: "1/1"

  84. {"7112352468224238429409922076","27352613969485479992489","3011336078611350743645389718","6278469788120133912214","17772001122364847982122","527545361454189926757560822","506995224502863919779463279865","43542983721426552973505","113669569184981959854560","583717699906686518981997668325751","855023285363777771805406","774283513464718025038431594","1255679242170974394082150041419290","756177791242189726033353","475166424338097541901032170"}

    79

    Returns: "16553154901/1307674368000"

  85. {"7112352468224238429409922076","27352613969485479992489","3011336078611350743645389718","6278469788120133912214","17772001122364847982122","527545361454189926757560822","506995224502863919779463279865","43542983721426552973505","113669569184981959854560","583717699906686518981997668325751","855023285363777771805406","774283513464718025038431594","1255679242170974394082150041419290","756177791242189726033353","66240156927285384121550495337035566"}

    79

    Returns: "12038653/951035904"

  86. {"31082640912240493333404090261941713","8477153491281313050053634","4260717196742755829436","9863358480936094621500628794","43775389238105586575863385","20405207910308207201038","5839572844268906353202","6638083241186610275682098283354073","5317506568408309525451","380716449395547043370281","8946850561117944306352","7629139344952000846676674186762","56705952114269705821513692392825420","6005942095150806526349037","19099860226698372270203602978"}

    85

    Returns: "10256312849/1307674368000"

  87. {"3108264091224049333340409026194171327477153491","8131305005363404","6071719674275582943","4986335848093609462150062879434377","48923810558657586338501040520","91030820720103805839572844268906353202","663808324118661027568209828335407305317506","6840830952545112807164493955","80433702810794685056111794","4063526762913934495200084667","674186762856705952114269705821513","92392825420050059420910806390911340","7575468116084","9526349037509099860226","98372270203602978148106707214955622"}

    55

    Returns: "509/28600"

  88. {"3108264091224049333340409026194171327477153491","8131305005363404","6071719674275582943","4986335848093609462150062879434377","48923810558657586338501040520","91030820720103805839572844268906353202","663808324118661027568209828335407305317506","6840830952545112807164493955","80433702810794685056111794","4063526762913934495200084667","674186762856705952114269705821513","92392825420050059420910806390911340","7575468116084","9526349037509099860226","98372270203602978148106707214955622"}

    83

    Returns: "3151053089/261534873600"

  89. {"406674359451387519375272134","393022780290948696412114094862256285620883","284969064764421822424637627203937356","4974251454934638100774066816763908053266598021","6714335","427199","70982256412809831240081","39498381434980380226553590502214132","31433726309342708900495495070094330739272496297850","713802900679778521869904239858186173504","87240729967555942267836333021716","88906615401200096279918995483541504193808119809","540443381550384532310550462127","9252559287477006684","47431183756149949758276943240712520"}

    4

    Returns: "2/5"

  90. {"340567256697708650874068480008447503995862","2757941736001403750575","1395279504383896","2859367079091346775006262025288167","969608682551964138536297341","477857359289882826774521522677920706831617","29512701258785294622212890487518474748135963235916","30309408921617702195113366018229256749767","6724953","17088655762055456062922260452877144884798889653284","9100089165079698848282332578505688181614604","6469514404239701001089911494207693","8228559783788698067180","45277046450194784391768497686115320840973438472480","2721718455161128878114780699080476320"}

    6

    Returns: "8/15"

  91. {"943338246265644725800173692","7016251066792771456331131872087335063","15478200170","2753806383937086149699436373592541140791587710271","528543149010444828","8035326707482547988244342968785663867398","4126439205223505754981656753477","215906036756924715197","26232825364541155955242003309041798714013137","43361847432040799","97223462834","802550099012658454516265337","5607219830418509188974286506592","93692746229735497563594\6969991115439080321525"}

    7

    Returns: "47152267/330220800"

  92. {"232898848834074972315487893659251","154008612478315225130719535060874141229239746","519105","3174342115795557418164863925351327321506636509","172116160255047475901148601861051","848458246369612109028098421116","26573460821838379649","42958861124","9427079423251152015977","806998034788259727904648619102320228691064","81226958","303038843543398115091721","69375673253110613994205146","95624312334534300906828869142776387094853153153333","3635219000452524791506758728381205762136433668"}

    29

    Returns: "22546402091/653837184000"

  93. {"232898848834074972315487893659251","154008612478315225130719535060874141229239746","519105","3174342115795557418164863925351327321506636509","172116160255047475901148601861051","848458246369612109028098421116","26573460821838379649","42958861124","9427079423251152015977","806998034788259727904648619102320228691064","81226958","303038843543398115091721","693756732531106399420546","95624312334534300906828869142776387094853153153333","299278629000452524791506758728381205762136433668"}

    30

    Returns: "0/1"

  94. {"232898848834074972315487893659251","154008612478315225130719535060874141229239746","519105","3174342115795557418164863925351327321506636509","172116160255047475901148601861051","848458246369612109028098421116","26573460821838379649","42958861124","9427079423251152015977","806998034788259727904648619102320228691064","81226958","303038843543398115091721","693756732531106399420546","95624312334534300906828869142776387094853153153333","299278629000452524791506758728381205762136433668"}

    31

    Returns: "42182982011/1307674368000"

  95. {"9304588208845291678","8744857770636124057045725","388563583341358312559231732520609","3398306008255738277678423818390606","604820132863086805273205101795855359","2811692756153542350741455328219030156423","7202414659298874950658883839163592844196","2757306351847070","81504802903928872890146","54314636343527551","5727928698152606626151643450168513398","8756853009178250885203741770","1699276889596920507546641369"}

    51

    Returns: "73255319/1245404160"

  96. {"9262002830492","50830941897341786409443","9021317367126019319355","18452406948427797667852008050284736462686602637","329162769925554614709524729645317160547650929092","29686746584014038","103006133892415132545873121216101249724698","17717113244667064470628918822159580632709","1084316518396454576228629056844118159058","98791280571950007251681722256215061135652505507","4795988119687466204929495613705407947071753395","62394362393667072844283072","763131399353151903521175395257120","58901516491747014630690039","657966273756334223227102"}

    70

    Returns: "44473879/4670265600"

  97. {"4795988119687466204929495613705407947071753395","62394362393667072844283072"}

    15

    Returns: "1/2"

  98. {"9853039964767099046848229711021102972577218","40430370757019136037655673764732185546222","40726536571808557405217785385"}

    17

    Returns: "1/6"

  99. {"788972404703700619573","2257584920685531038277","169538543557615","882028039817056"}

    66

    Returns: "1/8"

  100. {"178611350743645389","28062784697881201339122","510777","1011223648"}

    16

    Returns: "1/4"

  101. {"7112352468224238","294099220760173","3613969485479992489","3011336078611350","436453897180627846978812013"}

    46

    Returns: "1/30"

  102. {"711235246822423842940","92207601735261396948547999248942011336078","21350743645389718062784697881","113391221410","772001122364847982122342754536145","2899267575608225506995"}

    47

    Returns: "1/48"

  103. {"3108264091224049333340409026194171327477153","9128131305005363404260717","967427558294","649863358480936094621","1062879434377538923810520720103805839572844","689063532028663808","341186610275682098283","6407305317506568408309","354511280716449395547043370281","8946850561","279443063526762","239344952000846676674186762856705952114269705","31513692392825420050059420910806390911340","75754681160843","52634903750909986022669837227020360297814810"}

    95

    Returns: "1966376749/186810624000"

  104. {"123412341234123415451345134513455", "1837458907348905789137458134545154", "3485134513045034950349501394059103495045", "1893475983745891739857134541351345", "13908450810923183948569137846545", "190827348906129387469182736498712693874691287346", "1284016238974691287346784123412341234234234", "10892346876129387469128736498712364", "10289734089127349871023894709128730489127034", "120893460126346123908460128936489364", "12908374098127034897012893740981274087", "9823764987362978256978364523452345245", "20938645078926308956029836405896203986045896235", "20938645089162308956102893640981234234", "1029650891623059861029836058635" }

    34

    Returns: "641007001/23351328000"

  105. {"3", "2", "1" }

    2

    Returns: "1/3"

  106. {"13", "10129414190271203", "102", "102666818896", "1216", "1217", "1218", "101278001", "1000021412678412681" }

    21

    Returns: "5183/36288"

  107. {"11", "101", "1001", "10001", "100001" }

    10

    Returns: "0/1"

  108. {"70404111977878510001645675574577018815307462894528", "27996547821613825052279175191853827383057003930816", "46864115021536448229789388617887376852146803504208", "93859439197787520647533132638464224905325024260188", "56157120483948525932677523198594400206704630495560", "80884804914499628666049080937963228062645242185515", "40112308739094744183904664777856397876772362047682", "10576282835662378708613628216737469588865860854595", "44589919143592678917544519194198276618006501609902", "35778316031217329522614345158789262460601628074207", "54129385722270033574580200421238709422361088493814", "20344368007745184540899790536405436323465813285954", "11588575753898470105880537399092827552539067988149", "74668989719589264388661912707301101049985212680926", "85912994708642054429829665553158419737431153469600" }

    97

    Returns: "29629049/2874009600"

  109. {"4235234569872649876928734569123746987123649945", "6345634563456345897649876239487698234845434555", "2356678567246523452345234534563246356234562345", "2435643576436724512452356345634563456345634566", "1231234243634576467578689790698576346345636345", "8674567356234654357567879789068546734635635634", "5765467456745673563456345634563456345634563453", "6475976298746958726439578269487569238475603450", "9756763463562345234523452345234524523452345234", "7234578692348769287435692873456928374569926345", "5634563456345634563456456453563456685678678857", "2987562987345698726043502345987260498572049587", "5634563456309872093485709284375098720349870923", "2345234523453245234645674573456743754674567454" }

    97

    Returns: "81703987/7925299200"

  110. {"1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15" }

    99

    Returns: "0/1"

  111. {"5", "123", "21", "451236", "98413", "9879132165", "231656213651650320651968547613247", "65200013", "298132198465198985196849846516549874651684981651", "9846518941163516198465168572", "6516851987461650", "89849819897984984981984987498794989879441", "98416519687961684984196897984", "1321132132132132132163549643168794168498", "984651651901968497435461984964631984984981984" }

    24

    Returns: "0/1"

  112. {"1", "29", "97291", "9409", "88529281", "7837433594376961", "61425365346268570446197767595521", "912673", "832972004929", "8587340257", "73742412689492826049", "7153014030880804126753", "693842360995438000295041" }

    97

    Returns: "4903/60060"

  113. {"10000000000000000000000000000000000000000000000000", "10000000000000000000000000000000000000000000000001", "10000000000000000000000000000000000000000000000002", "10000000000000000000000000000000000000000000000003", "10000000000000000000000000000000000000000000000004", "10000000000000000000000000000000000000000000000005", "10000000000000000000000000000000000000000000000006", "10000000000000000000000000000000000000000000000007", "10000000000000000000000000000000000000000000000008", "10000000000000000000000000000000000000000000000009", "10000000000000000000000000000000000000000000000010", "10000000000000000000000000000000000000000000000011", "10000000000000000000000000000000000000000000000012", "10000000000000000000000000000000000000000000000013", "10000000000000000000000000000000000000000000000014" }

    97

    Returns: "32097931/3113510400"

  114. {"28311274336383644215941967695364832586894742880440", "59419354256203177270870036853962942235659509731131", "93950621045280882864557306505793292519191095671008", "56521736733192358137663353772971298324195851390842", "29751876712597372905370336895441347889693219779492", "91236016016905011102140734270245475476591538409647", "85684623232894153332655626811204386128157216283630", "37122235197616449850738844847524732890607282012780", "21025377773442562455868574038884217046200732186203", "20151232546618575694962638525594515025605043068983", "24923598999990256874104733969187651962501868095554", "42325708145136479502217057783632289531632347843450", "89118426059559050014308492643913519392916341726309", "30459206286864476966770541810119599138660823445153", "28324294379058649790812514407834404754916935261543" }

    3

    Returns: "0/1"

  115. {"123123", "12313", "33333", "343434", "4544545", "344334", "34333", "1111", "2222", "211323", "2342324", "12312333", "23421134", "2342234", "2314234" }

    28

    Returns: "284567/29937600"

  116. {"4781390412287981354", "6971388200062", "71903903832", "47813903832", "4759490390062", "47813901058", "478139027223051", "475948819758", "4781388222254", "478138822167", "41243903832", "4816903832", "4781390217928", "26681390248077", "47813904116262" }

    73

    Returns: "1/1"

  117. {"9", "10", "12", "13", "14", "15", "16", "17", "18", "19", "20", "21", "27", "23", "24" }

    11

    Returns: "9617/120120"

  118. {"91919191", "91", "9191", "9", "1", "9191919191", "9191919191919191919191919191", "191", "92", "919191", "991", "9191919", "131453", "194954161697113" }

    91

    Returns: "25021/9979200"

  119. {"99999999999999999999999999999999999999999999", "888837477775777773888888888888283", "9999999983948139241382461837461324124124", "412341234123417669813264798172341637845361481", "41234123748179234136715832647813", "23141236123894612378461237894153763481234", "1234123784137409957234095872039465789234523", "234174183924619783246198234618934689173467891234", "75246523789457189324082361487916783246198", "41348791832461837946189237461783457183413412", "4123740758589613489745324789641237846127893", "413248713874619837413578345347856783214", "1234718905234705627893456198641892346895698415", "31287490137546238795618923469817235689511423421", "8942757892346579132471283689658916345" }

    58

    Returns: "6012420581/435891456000"

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