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Product of 100 factorials (Posted on 2017-09-07) Difficulty: 4 of 5
Show that (1!*2!*3!*4!* ... 100!)/50! is a square number.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
bringing my original table up to n=100 | Comment 7 of 8 |
(In reply to when it works -- computer exploration by Charlie)


In Visual Basic the numbers can be approximated:

 lsf = 0
 f = 1
 For i = 1 To 100
   f = f * i
   If i <> 50 Then lsf = lsf + Log(f)
 Next i
 Text1.Text = Text1.Text & lsf / Log(10)
 cl = lsf / Log(10)
 Text1.Text = Text1.Text & crlf & 10 ^ (cl - Int(cl))
 
finds

6875.94879958824
8.88790877214531

meaning the result of the division is 8.8879... * 10^6875

While this number is beyond the reach of UBASIC, Mintoris Basic, running on Android is able to handle such large numbers:

BigMaxDigits 7000
Open 2,"prod of 100 factorials.txt","w"

sf$="1"
for i=1 to 100
  sf$=BigMult$(sf$,BigFact$(i))
next
sfp$=sf$
do while len(sfp$)>0

  if len(sfp$)>80 Then  
   Writeln 2,left$(sfp$,80)
   sfp$=Mid$(sfp$,81)
  else
  Writeln 2,sfp$
  sfp$=""
  endif
loop
Writeln 2,"  "
print sf$
print
a$=BigDiv$(sf$,BigFact$(50))
print a$


sfp$=a$
do while len(sfp$)>0
   if len(sfp$)>80 Then  
   Writeln 2,left$(sfp$,80)
   sfp$=Mid$(sfp$,81)
  else
  Writeln 2,sfp$
  sfp$=""
  endif
loop
Writeln 2,"  "
print BigSqrt$(a$)

sfp$=BigSqrt$(a$)
do while len(sfp$)>0
  if len(sfp$)>80 Then  
   Writeln 2,left$(sfp$,80)
   sfp$=Mid$(sfp$,81)
  else
  Writeln 2,sfp$
  sfp$=""
  endif
loop
Writeln 2,"  "
print
sf$="1"
for i=1 to 100
 if i<>50 then
  sf$=BigMult$(sf$,BigFact$(i))
 endif
next
print sf$


Close 2
print len(sf$)

finds

Product of all the factorials:

27031768576517499710558090027596232298558093274311372898878080715359180165112031
92430745615846524837281877965192519547242721319043018325169310301931868967540643
14636336565956356312445318234895459814650063457694775386003534187336458680650104
08042273812281857257966951771254299784907532174172675261042904443735033006827062
36778173493240290638006467655691485330011773845402913858411465364146099987929711
07248285578361460512735635185296745657732007210556786440256146238834812375648946
28859759852397473591728651044381377039700592548741212705965716337933203079627369
26209714294067025831911541041901708864208596321666458768752855221418838604709774
83413478260824926725758719163597242888329574231309601634465772012970300810368195
56978682525116845551863481353447203913609105954470434446731819716372893950027080
53776526038788649748707391188333733588816329612086059297012947441482071618624236
11135988518774306790531853022907113671006921381193655357496123978122118708954213
00661971739287030299385145794712666816403151658293555299062759180558560552120065
73335985642451371095025560051530906923476102408031508391293688005954319419696700
27750573911439325191780646449295619341707000177257947490657835366048695329722167
29771869515017257460764744205293736768850910844817317789760525673651462286969218
05191278304080317350286859334301998665192886284986617130635590172759039508366671
07086342982683979914172377198293527050000187166988831711208485886557984561887523
17306393075998430051287601872760427292416532717472158265481963073016090468249993
68688213299225131493184437781262532487305838868519552740773789186747335272711155
03361964542028872999048479776540103296507470121421437410170274533761468313384556
08413252873927221734647038025714766776152183177488678194636943158142340296086335
99174191164166761332146194363811254653942551220731704520312562726502005923544842
26488412926464250062010441856848621055747269953050315075415493705687224027918534
20112332280884916208713251458554365777660881201931186067005132165233655598425788
27753350118214224351849231891524367465109857971730903623200901737727610030131893
99204885905573045326790744617557179216123899953440718447397032345719199566328080
14778170148649586922770228743618307005560442586998801940863559279122530838019077
28033576990057498767073563782287469258373472899299682869394734246293245657368952
51504236909894050964735769846859410210914361235760266749562663433119996864692506
83711889792780188010000923545083693616512431299553430423319151799092729329012624
79194240848255121642293665462038958840930785647155325545917420948854172079772476
43928789402951531251394269178443480143247171017122399482759113314513506255030799
13906081322162453194924075795374300445602930241718494950581280344647070513379468
66001874966676677933798271430791857849065185862315553175343634193923031371621108
82056537588038562114245119424461786161239196321586812093407662648436079181907111
95995769497888074300126090980893407856203416039961905820434437338769169809988554
94051498494077188351066933592215965226094349540391408233590165480330071635742331
10076720709478569376146123344025455777316135455888751337560280022533926125250178
82998610886274897113627550258742450493613087049732151945364827840949182968966809
15760065909942083295976297734855640033279573335766887567170128208932374855044247
64124069323911108820627399626525024382875010796885736339986584989320120423860864
26239238190775121738551066893598665935774455519152157588419955426909632953255225
94414754686692674264643003599275535383376806354178268425133780941960260847711464
35066774964799840510519597652773438152490596634012188249819523969133086597657206
98334997196469642114605784797650509426457220495304067174165545766010632963921930
39492327562587891925104770131243101602386778121090770547039106437549341643107353
72104825239197655045027533293444926202961463847359407615889974303054554470962079
37391309074448688892666491670891709155822800892901929566779139684433502227476274
01951537089488766480892968651867475549208869007056385546152229481814724380376495
40053669840530068127131126032909944816787535722142055394929041165731570535386655
42029486197003757482662830062120635456960276391311596048447813899284882862556031
29826479768237355662844461964265332432885676789484830341406616197694815544922098
80758513751326398774755109237791432074048823410988991352369647941838207035070662
67800973792432111245784381086462014399154907268442765473783291278952434015025195
69958065946570342607659633790262993423968134302796490007265371032283229222743016
18581208301765561446392193493904232903108740309423206747998548157751484010320114
14853381341522616342671480892280541482672772691715237914427732529513973227685560
10242419477530056566052252015919704031525621127308176552011886893502395670009792
55537747478112682709916534868954877256632775752861030509773615848960062886356692
10151264444972709370572481075509070029875071185160529111621252142202673338305550
36019207093303065134403780316668192566918466894459285904966942814743846131301960
72910335628851658625694790272152049387982349837780381922503577826702544753164886
75039486787835966235572450777593617425795005589371791584337812875284631159728936
75648877895421050760033131115149405276115595745764609200397590716143404695331463
67249323666949235866009567675661746881544987555182401313308529277884051083168608
39524389146592070352766013263554814855316560483930269161109555115598599074606380
41934670628168712275613923627233487463651574284673242624918929529412607718318038
65376280705773044319551136726449649081919669097643839158980839673527659071144056
25567363146867872227797114185869379776838512786074844136826333346246190407373179
25896307836174995601669937045540333703699608766332685598683035025973316356315674
89539184136463607408207731355177015776171295118476659007517798131708995179264232
97750920427028974700540224251803798780700829498972359884800000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000

After division by 50!
  
88879087721723246827463836488541525566186766028907558970189701680165147413910850
68857234440367519707696389797763167693938266400593372020366188861001915565075431
79203072473263746969745761495898941222173496085179328042282863957032277336655035
40308020010409096460166742266573555347650251089682790510729316282129862881323761
68536977023213072512963790997069563444583603265996814606920646835260461413215245
04153337056549576909595295015188480191226589603529005113464039489605248367282993
12973576159766299549230160242738189241703425741011803240478161236998606565750639
47594321109678118750653489230669883359088548477945429363791274468019589542234470
13969328503881523013076428970527001562856649526153227676245284223079551567809988
65391707370140903477179641355251146757656328625169778102466564585495095792341083
53760324408182752359274874253871635268417849764885283643785502933946304427357289
68325282882474977831999164203029009172925587035972162910299082693664522878053538
79470671272280181520522978742074541472745336785404225148564962320029636133774148
11806321304340482314855665154464077067174506040235043265101061826679718939692900
32334160300010844762730669750319490994122129525396878655262477548873333564181112
88131481636777043265830936830774059284142321281806698400369772260648738666690282
53177354045414646008215661744534820838727716695459860114238640110324389229905780
72170616797978918918702100032169926544891624228329126774678326025911694607900853
69764517978019534121423037448769017836494254888806667768373934206215811210146901
69012906096651944622669594256212994288171427852016232667010603374536910172473836
89374386309081124347151606816082617562216700396258706317586670416036268374339136
62081221034780564589657082300718680299923428963169828590296406337712774771003605
58304307638948303907488042382002159891244735612502639189915706136886553419339262
40251895160731907693739235588752291923123496187029918061595529318738761054456862
67099395634561460580174707592572679570069840376579507324427501408103905556617595
48104804470312464258601583333606031067945000754906893230106334915504792331574346
01426811123757521734189053769992802535491025392513971613607567236778318422167831
06665604630811132968974455752622878675464586320590872119097875777566803517028734
93970225709245281987961953796301970323962659217205128929378397335798352345678350
43577756735190254811821028021547373456308998495281365803581174824648700297037115
40272495651161976236443759272297942838117791477371971740042608469481410639762188
07963406405944514216612979536046127145194684312633351637287909440109024774830398
53479527575118388974390785931928835895792137293465518417869266633473032809691892
10816053310907019835755133403959751567784490382727531235627109936212255346132011
50870426823967278903428163246332382111704780482474892204244928575382301578060304
57630247264083887193442276905595224456019044621313232496608547402887100315029739
73103986542165191550796393276664723718074442121617367064332323939167764226453888
70259303459405500159435618165868134088817964974415047387717454700557754786763212
76468213523653745446686902232286581984743721646515072644466846971966024657523140
49584360406955126738930167676334420583994245134807849032956015687967687447501902
32537021866932383431513933944578600869053049712664849648317675221632315748877197
75186420111214377797289753793695627627387949750829204409557390396800193696602451
25163587086658651074820575434355147828408642114761763171573320754313451066864083
04343101883443754454987727275527231989039768688120279470939796458432468413899984
55502230037655563938229750099089380290012280310272009545253095812419824536499677
08865399533457907993668939970502628451745316758375619277271968514795829657040406
76839321409178564148520773480982016538219171219954492666997055996122100521594187
00288369315457264372839115334593757883885037980836828334675892978688324928795968
13836690782169500724098367472327629184919789512908119828769218306076354008829747
70454445092816693022689015601519130764379655938004104272850998625346437282363814
83309049813859259857998996333777407499192342163614448268709400667523922132557141
06920155795909460333721040566222077421812568594666448805020095407437081777071624
03498819462825815151997568900968568649303097295940273703926436967261048745811620
16816063629447224882102277571691204999988134669941792468924498958932297772918022
99971307701598550540897371456446168155476336053190027579698082871890155424322166
55942484895628503066089928936430729837032397146534698282301808063377526388124099
92995866896137033529492918557839428732625705474520202730021863856431750090420891
72774007426401491187785981102039235093350502912934210831968437193731748617657855
23714179050482980800966972337431061594447453543179801899621666645050489126645274
02077562116024448266493981915747118851125397466906380988673501712827882760896089
41208398897918781941366285262365371059460824641658780592792583696817727503820237
76475231506213181441925236535198303001564660584025822729703206026220289463387790
80223538581200392158588865526614815262946550822203998047333750971678639141014046
40379861734997056488793224600055681448457164316283944168462066591039090809792059
70190106893422213085083864592143107257368158383505771318247118637397851909029601
14723940237326466674896228417213014947058093519681872537562824970083125582688222
41206233567687164441533977445681291791565778494847929243115477656815374088472272
14826150360234022487427700663725559255332960494713980379459534731977321598235249
51840301265137216449511305671636690269709961293687062942170925618251288416477996
59664236102186267575772103503803630610196058922100356804797444505066404569036454
72110847347238280553896373944398875577165261451345354619813762882195049916270425
27958885342678360135304537149876127549406107493517055451072329503894914665766348
39040000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000

The square root of that number:
  
94275706161090751507053966945348267315040284775057725249817264189213946654708491
71137849217623408572040505741221915976444154504978093659655246756532288692385795
57859765992252980283292977504307941936196098206094050090375649205033805536047220
63612260169099821214890898130471170839972936097180381165711546704781557416753870
83633412339569219319396604123068370649232501003092916405425452198771934786242303
47297219014889900711709621893667867404164894731691589419071491582145951270320308
16144452761497188514911234900852483175407119086089762243193408515146215637139611
29719187618193126441188084028414806902744820687297701208683972093095545811475251
77292184871211627628077331212287600604130670381138162304315192596355801473452825
87979817135405133202066028758008995915443775892370216333614634365040712161401835
55596230676893831592465882724438277948209581602386355366719702742991466375500383
51474460746820858683421753985548523006772394709153824523871370376984039939115763
87963947166320609261307838875441650085461330992417956653776419606323723626170731
86597466661964083843368878461046772575757738914916242897902280182883607734020673
71425703261605968044076139674977974593696793630996867563589243094228260003683545
70718198112844574004662869334166804551225487656666906005411402934238834219962053
79586728655993060009237858461485924003069896615615844878026043024068678705271177
75419967557162904030022401036227582335078995520925921504146054015138953841869638
95151703092830777236024496874188519684097118897039997333375002233988006588365328
09430158417533552994671587178092807305165796717254548194589249805497012989371287
08043714712286078719258729807396900582701897613839353399413178666970752310762594
99253845576297913465821590176034261582136163913162045832763101585667036319908514
59655273403329904021275208052876136307956108489993397392568293665277738645071730
62082489835485733367246494345080457313899867472478390094877603404811617655690822
78801839988932778286540852369148975498687520723000309925612159750886853795584412
54571109469688029209791991772046036311739895211764401880939286167314722674127177
85430983248612745226378366977127666163338032114431055492287791141315688758732128
70587307114333244054730262929359376113852839256172009728170521496236102937512002
94684802220817890039084279643437467553902534726719321424132691220778488336090966
26199290213038232081477998488452321268538526055938930048781116568621896345950119
59035980842929966297938376185556769673278572007990991979543719611265562408909439
02408115936537409171134866898155083613733592092947264583637530297755345954994348
00245115891337124546715812914241640534873738180002228654619685227079206712471470
07137725605576101736929220017144588193138719159609246063430247112437339526067780
04182052192808576999989351505168726922320179126383136534789601371371518945655355
86619618346971671203772393970202884212689480815776291370881482513290761569063403
52000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
  

Edited on September 8, 2017, 11:19 am
  Posted by Charlie on 2017-09-08 11:05:02

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