Help me on this sort of question

Method 1: 

To find the unit digit of the expression, we need to find a pattern. 

After writing out a few more numbers, we see:

\(1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10...\cdot2012\cdot2013\)

Since multiplying any integer with 10, the result ends in 0,

the unit digit of this expression must be zero. 

Method 2:

Of course, you could multiply the whole expression and calculate the value of 2013!. 

2013!

= 2842867247059648714320882908184358975298425936765406743563017882090099387783580447012935263746562

335479064550804211732063022834149625421992014144514533688428601587813569194324989561917151782797873

121680666949119475158973400795970620477769770891025873157955892369624843826017526081184552011934374

10509731578204812073608581513295488735471553922625629140543458920383377989900058870151031893904324

786467813499293437953323054452848703591411894569362652829036859143720665327952287750921278821073795

758246971107141102165987055971413647452259971850140909083491892667926811689827067309820345136189931

70855874333269695793623349197860089943442174193730066430857215683894230520597646816671734574453822

639829410734166055686591247872808311907999627073361328419227145279682743490293656700038542243000003

54656476434833237074182399956043794238102817592695696938568409658563414032484092565521956662668178

04130797077932032476566168993055491553005065129612241809221756958758551301616031692650277494508565

460020142819737041175520752950885102631597178638938715655511417824993651493965622682149873599776821

16903361743064161466851269709092940831385528741007252712667479425921896880812303322531002514033730

27228801998127490992546175714650004281060557576018602691444874370276588961454729805801934914327058

639037910641855479898434777131761105855951851430766729127680477612727441963181686830328681323514755

07005255980785550480809320586257442330649968338872970362487582453860064408263458172262252063541389

40472165440197844386043987814485357207839608993158387923679839401916385370678987012450565846378062

37613648468018169610742424883398829055732890553015662215573197318332952206241281889197604097179798

088111804842744903817885667786672979352296770176797979680935970290662288862203487507979401519096847

866678683866374334376409418963307236435361004707445941188594507325969642135133107415655296396624159

611926210964204186787681300922165759386829935051758089290240223443538483111578632645575815706664545

129018761043826893436452374686888469782790455363052163042518632450978515011042593378534376399702189

20273056745879634273623885977170895848007403898018718056610467971822515583288587585365491735918541

851143874245054399927740846012278308508172250269186018235162874294246905205271515797568854333409581

770621535680031307344634119307920989359990306607019552656310949483911595447438683550126438222044423

66574187301773842976392960318817492317270290321216739518314342760193902063057941457483831918266124

856180712708873479072562585163511439010136904713443685275196453218976196831564862256983178474297470

717874357825058000841842254607086820958709964285390681344929611021650748906543867856832645308948315

63132549697850639084675795153902191945156201050737533340297121449992322830373504638815285378819787

554773575857714346271171627666997528270237625481717740484365952887157175985255192895785552274978335

022571528293668119818363728825071916771945692681017826150901126099177461980958354110093788032189674

99188857015469067059756448734088835527244682814249255050856850527306308568919331798552245212702085

62951496568791453972719099564639553188855510265294703433263335170190878675189768260844753417897774

605623321349011902193140551488084053156588942044109445461823324089443588958669455041738856562689822

933876705027721348392092785430536231155624184547493347003183551786284141683397325217471718551661094

77719366226966982601829031348339568014756885616782225353731679589010642963026728648636517643953417

22332329923130642392597499850698224808818858610577949452559361090645680123768915090996283714284645

18499556483198459740683706440681874790291565697629012736855369437572800033629881008632192695545890

165897605607139837425754932756265266177004015134335575298431975102791240204949642576808809116443446

03495723958279131978382862453869345658201667502410202572628757335499674300863432592256152060213732

58766077086853543964535758107538070039372783281779176173268921617552162373199770869845016818789242

170921555048050613527266247691008484193008401103135422070066418155209427253547525169099964052804460

89149387826652320289706871854037767039903259795085325094930026669064944815410134739486230574390677

16354970459715223485500393463339996144504094685302690606100003782316415853025878871684344019606419

775272528889669587663352056426642112160119700550423526808274366819908984680988815165676455989729650

20594215802444175390346271358292514861528156681872687363261748664778410513181823674285103194306067

045099545803147235396638706228707189288952637962203047115037086421526867398070632958542701199245133

382396133624499203582231312010840884212052056354447679961154306001820509943943259757985331301915430

07073726382576912304230842839818060193895673180137969400931470686507837694128076839580668577787352

528579884315928111505284991546171783935459962376675014391494104156202279442380953797629681894096893

415368905476382033883187685926067674083948155038913063463236521164557031729533512019335884749555275

63220129086963377342371823922961841428147215856281545446983868005765000965710748347784855863778139

511179159754686588299301850513114088473086011993200873704826365083841064351095378526835498963203426

65003300891762896539159219941617766093956216205158201836090300940905254000256692488424335980452653

96159843083730318095959670247829067563318797906385452400640000000000000000000000000000000000000000

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000.

Indeed, the result ends in a zero, a lot of them. 

I hope this helped,

Gavin. 

Note that  1 x 2 X 3 X 4 X 5   =  120

So...any  further multiplications, i.e., 6 x 7 x 8 x.....x 2102 x 2013  will also  end in  "0"