Day of the Week for Any Date 1

6/17/2015 GreyMatters:MentalGym:DayoftheWeekForAnyDate(Revised)

http://gmmentalgym.blogspot.com/2011/03/dayofweekforanydaterevised.html#ndatebasics 1/14

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DayoftheWeekForAnyDate(Revised)PublishedonTuesday,March01,2011incalendar,feats,memory,mentalmath

Note:Ifyou'reinterestedincalendarcalculations,youalsomightwanttocheckoutmyQuickCalendarMonthCreationtutorial.

ReIntroductionInthisfeat,someonegivesyouadate,andyouquicklystatethedayoftheweekonwhichitfell.Thisnewapproachisupdatedforthe21stcentury,andemploysnewtipsandtricksthathelpmakethisfeatsimplertolearnandquickertoperform.

Approach:TheDayoftheWeekForAnyDatefeatcombinesbothmemoryandmentalmath.Arelativelysimplemasteryofboth,though,willcreatearesponsefaroutofproportiontotherequiredwork.

BeforeIdescribethebasicsoftheapproach,I'dliketohelpyougetagoodideaofyourgoal,aswellaswhat'spossible,byseeingthisfeatperformedbyvariouspeopleinthefollowingvideos:

Dr.ArthurBenjaminperformingatTED8yearoldFlorianOsmanionGermanTV(inGerman)

Here'sthebasicprinciples,brokendownintosimplesteps:

1)DayandMonthsNumberConversion:Toworkoutthedaysoftheweekmentally,weneedtoconvertthemintonumbers.We'llalsoneedtoconvertthemonthsintonumbers,toadjustfortheireffects.Thesearetaughtinaneasytoremembermanner.

2)Additionof3Numbers:Withoutusingacalculator,canyoutellmewhat6+6+31is?That'saboutasdifficultasthebasicformulagets.Ifyou'recomfortabledoingthat,youwon'thaveaproblemworkingthroughtheformula.

3)SubtractingMultiplesof7:Let'ssayyou'reaskedaboutthe27thofamonth.Regardlessofthemonthoryear,wecanstatewithcertaintythatthe27thofamonthwillfallonthe6th(sinceit's3weeks,or21days,earlier).Sinceadding6issimplerthanadding27,andwillgivethesameresult,whynotuse6?Ifyoulearntosubtractmultiplesof7,thismakesthearithmeticsoeasythatyouwon'thavetoworryaboutadditionproblemsanytougherthan6+6+6!

4)YearNumberConversion:Afterbecomingcomfortablewithalloftheabovewhengivendatesintheyears2000to2003,you'lllearnhowtorememberandadjusttheleapyearsinthe21stcenturytokeydates.Afterlearningthose,you'lllearnasimplewaytoadjustforanyyearfrom2000to2099,andevenadjustforothercenturies!

Runningthroughalltheseprinciples,therewillbeanemphasisonrecognizingand

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Basics Formula LeapYears AllYears 1900s+ Tips

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takingadvantageofpatterns.Thequickeryoucanrecognizeapattern,thequickerwillbeyourcalculation.

We'llstartwiththecodesforthedaysoftheweek,sincethatisourgoal.Alltheformulasandpatternsyou'lllearnlaterwillresultinanumberfrom0to6.Thisnumberisturnedintoaweekdayasfollows:

Tohelpgetyoucomfortablewithconvertingnumbersintodays,taketheWeekdayCodesquizhere.Onceyoucangetaperfectscoreinashorttime,thencontinuewiththistutorial.

Next,youneedtolearnthecodesforthemonths.Liketheweekdays,theyrangefrom0to6:

TaketheMonthCodesquizheretohelpreinforcethemnemonics.

Note:It'simportanttonotethat,inleapyears,Januaryreducesbyoneto5,andFebruaryreducesbyoneto1.Theotheryearsdon'tchangeinleapyears.Leapyearswillbediscussedinmoredetaillater.

Onceyou'recomfortablewithboththemonthandweekdaycodes,you'rereadytostartcalculatingyourfirstdates.Inthenexttab,you'lllearnhowtousethesecodestoworkoutdatesfortheyears2000to2003.

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DayofWeek Number Mnemonic

Sunday 0 SUNday=NONEday

Monday 1 MONday=ONEday

Tuesday 2 TWOSday

Wednesday 3 ThreefingerslooklikeaW

Thursday 4 FOURSday

Friday 5 FIVEday

Saturday 6 SIXturday

Month Number Mnemonic

January 6 WINTERhas6letters

February 2 Februaryis2ndmonth

March 2 March2thebeat.

April 5 APRILhas5letters(&FOOLS!)

May 0 MAY0

June 3 JuneBUG(BUGhas3letters)

July 5 FIVERworks

August 1 A1SteakSauceatpicnic

September 4 FALLhas4letters

October 6 SIXortreat!

November 2 2legson2rkey

December 4 LAST(orXMAS)has4letters



QuickCalendarMonthCreationQuiz

MoonPhaseForAnyDate

EasterDateForAGivenYear

EasterDateForAGivenYearQuiz

POSTDETAILS

PostedbyPiGuyonTuesday,March1,2011Labels:calendar,feats,memory,mentalmath

73RESPONSETODAYOFTHEWEEKFORANYDATE(REVISED)

BramJune19,2011at1:47PM

Hello,

Greatexplanation(asalways).Smalltypoonthefirsttab:"Theotheryearsdon'tchangeinleapyears">theothermonths.

Bram

SapphireAceOctober11,2011at3:12PM

Thankyousomuch!Itwassoworthit!Andcouldn'thavefoundabetterteacher!XD

CaptainNegatoryOctober11,2011at4:34PM

Thedifficultpartisfindingthenearestleapyear.Orisitjustmethat'smissingsomething?

GabrielOctober11,2011at6:14PM

Thiscommenthasbeenremovedbytheauthor.

GabrielOctober11,2011at6:19PM

InPortuguese,thedaysoftheweekareordinalnumbers,soit'seveneasierformetoapplytherules.Thisisgreat,thankyou!

radicalpiOctober11,2011at9:57PM

MaybeI'mdoingitwrong,butNovember18,1969doesn'tseemtowork.2+4+6+1=13,whichmakesita6,soitshouldbeSaturday?But,insteaditisTuesday?




@radicalpi

November=2Day=18Year=2



1900's=+1

Explanation:

68isthenearestleapyear,minus56(6856=12)since12goesto2012,orthemultipleof12ruleitactuallyequals1.Plusthedistancefromyouryearof1969(1+1)makestheyearcode2.

SoNovember+18+1969=XOr2+18+2=2222/7(Remembertoremovethe7multiples)=21remainderONE.Now,becauseitisinthe1900'sweadd1tothedaycodegivingusTWOforTWOsday.So,November18th,1969isaTuesday.

Ichoosenottoworryaboutthe7multiplesinthedaycode.Itmakesnodifferenceifyoudoitwiththedaycode,oraddthemalltogetherandthendothemultipleof7,soIdomultipleslastandjustaddeverythingstraightup.Itmakeseasiermemorizationofallthenumberjumboinmyhead.

Hopethathelpedyou,ittookmeafewtriestolearnthis.Pleaseletknowifithelpedandifnotilltrytoreexplain!

BestofLuck,CaptainNegatory

schmozerbeastOctober12,2011at12:11AM

Myquestioniswhat'sthemethodbehindthemonthcode?Ratherthanmemorizethem(andforget),I'dprefertobeabletofigureitout(andforgetthemethod).

Afterlookingatthefirst7leapyearcodes,Ifounditeasiertodividetheyearby4,thenmultiplythatnumberby5togettheyearcode.Forexample,2068isaleapyear.68/4=17,17*5=85.Ifyouremovethemultiplesof7youget1,whichistheyearcode.Oryoucouldjustaddthe85inanddothemultiplesattheend.

schmozerbeastOctober12,2011at1:11AM

Also,the'00leapyearexceptiondoesn'tseemtoapplywhenfindingyearcodes.Forinstance,tofindthecodefor1703,youwouldusethecodefor1700(0)andadd3,eventhough1700isn'tdivisibleby400andthusisn'tactuallyaleapyear.

AnilJadhavOctober12,2011at2:55AM

Amazingstuff..thanksfortheexplanation

calicocalOctober12,2011at8:42AM

Forthe1969exampleIgetthat68isthenearestleapyear,butwheredoesthe56comefrom?

CaptainNegatoryOctober12,2011at11:47AM


CaptainNegatoryOctober12,2011at11:49AM

@Calicocal

Regardlessofwhatcenturyyouarein,whenyoudoayearthatis5680(orsomethingclose,hesaystheexactpartunderleapyeartab)Yousubtract56,for8499yousubtract84andfor2852yousubtract28.Doesthatmakesense?

@schmozerbeastI'mnottheauthorofthispost,butI'mnotsurewhatyou'resaying.1700isaleapyear.

01e402489f8a11e08a9a000bcdca4d7a



October12,2011at3:03PM

https://www.facebook.com/theHlarityensues

DanOctober12,2011at11:15PM

TRYTHIS

EGDATE:16/5/2020

AddtheMod7oftheyearspast2000[20]mod7=6

totheroundingdownofyearspast2000/420/4=5

tovalueofthecorrespondingmonthvaluefromthefollowingtableJan=1|Feb=4|Mar=4Apr=7|May=2|Jun=5Jul=7|Aug=3|Sep=6Oct=1|Nov=4|Dec=6

thereforeMay=2

and

6+5+2=13

Now,addthattothedayminus113+161=28

andfinallymod7theanswer[28]mod7=0

0=Sat1=Sun2=Mon3=Tue4=Wed5=Thu6=Fri

Theonlytimethisdoesn'tworkisbetweenJan1sttoFeb29thofanyleapyear...Thesedaysaresimplythedaybeforewhatiscalculated,EG/SaturdaywillbecomeFriday.

Tryisout,andletmeknowifyoucanbreakit.

Dan

DaddyOctober15,2011at11:50PM

Forgetmemorizingthewholedamncentury!thepatternmakesiteasytocalculate.

Simplydividetheyearby4andaddthat(withouttheremainder)totheyear,thensubtractthenearestmultipleof7andyouhavetheyearcodeforthe'00's.

Forexample2025:25/4=66+25=313128=33isthedatecodefor2025

DaddyOctober16,2011at12:16AM

@schmmozer

withregardtothemonthcodes,IamlikeyouIwanttoknowhowtoderiveitratherthanmemorizeitIdon'tknowwhyitstartswith6butthepatternissimpleandratherthanmemorizeall12,Ihavesimplymemorizedjanuaryandjulyandcalculatefromthere.



thedifferencesbetweenthemonthcodesaresimplythenumberofdaysbetweenthatmonthandthepreviousminusthenearestmultof7.Februaryissimply6(january)plusthenumberofdaysinJanuaryminus28.6+3128=97=2

Ifinditeasytosimplyaddthenumberofdaysover28ineachofthemonthstothe6forjanuaryor5forjulytogetthemonthcode.forexamplenovember:5(july)+3(julyhas31days)+3(august)+2(sept)+3(oct)=16(14)=2themonthcodefornov

WayneMarch17,2012at11:24PM

GreatarticleTwotips.IfindthemonthcodeeasiertorememberaspatternifyoustartwithspringmonthMarch.Thenit's250351462462

Howtofigurefirstdayofmonthdateoffirstdayofthemonthisthemonthcodeplusyearcodeplus1(minus1forJanuaryandFebruaryinleapyearsonly).ExampleMarch2012Yearcode1+Monthcode2+1=4,daycodeoffirstdayinMarch,ThursdayApril20121+5+1=7/0,daycodeoffirstdayinApril,Sunday.May20121+0+1=2firstdayinMay,TuesdayButJan20121+6=7/0firstdayinJanuary,Sunday

Tryit.Itworksfor2013.Etc.

kristiJuly25,2012at10:28AM

hey,iseemtobestumpedbythisone:OnwhichdayoftheweekisJanuary14,1900?YourAnswer:SundayCorrectanswer(s):Saturday

January=6Day=0Year=01900s=1and1900isnotaleapyear.

thesesitesbackmeup:http://www.dayoftheweek.org/?m=January&d=14&y=1900&go=Gohttp://en.wikipedia.org/wiki/January_1900

errorinthequiz?orhelpmeseewhati'mdoingwrong.

thanks!

KantheshaMurthySBAugust2,2012at4:20AM

HiMynameiskantheshIamhelpingmysontotelladaybutaccordingtoformulamonthcode+date+yearcode=weekcodeoctober7thof1974iscommingwrongmonthcodeis6.dateis7yearcodeis16+7+1=00isSundayactuallycorrectanswerisMondayCananyhelpmeplease.

kristi



August2,2012at9:30AM

Kanthesha,

October=6Day=7Year=11900s=1

Youforgottoadd1toaccountforitbeinginthe1900sversusthe2000s.

(http://gmmentalgym.blogspot.com/2011/03/dayofweekforanydaterevised.html?showComment=1343237285090#ndate1900s)

hopethathelps!

KantheshaMurthySBAugust2,2012at9:44PM

Oho!SoniceofyouKristi,

Iamverythankfultoyou.

thankyouverymuch.

GeoffreyHaleOctober17,2012at2:10PM

Thisisincredible!You'rewonderful!Somuchfun,thankyou!

study29December9,2012at1:11PM

Cansomeonetellmehowtheyarrivedat2091beingyearcode1usingtheirwaytocomputeyearcodes(thisisinthequiz,Dates:2000to2099)?Ithoughtitwouldhavebeen0.

UnknownDecember17,2012at1:13PM

@study29

Howwoulditendupbeing0?

208884=2004(yearcode5)

20912088=3

5+3=8

87=1

RanterbobMarch10,2013at6:25PM

Howmightonecalculatedaysoftheweekpriorto1600andafter2399?

SorenBjornstadMarch10,2013at8:28PM

After2399thepatternofcenturycodesshouldjustrepeat.Asmentionedhere,priorto1600theGregoriancalendarwasnotreallyinuse,sothistrickdoesn'tworkforthere.

CMSCollegeBefore80'sApril10,2013at6:43PM


CMSCollegeBefore80'sApril11,2013at5:51PM




pushpinderMay13,2013at12:16PM

excellentstuff,,,,gudexplanation,,,keepitup!!

KennCarlsonDyMay25,2013at11:18PM

help!July231991

soin1991:for91thecodeis5for19001999thecodeis1so1991=6

forJulycodeis5

23is23

5+6+23=34>reduceto6makingitsaturday,butthequizheresaysitstuesday..

sachintanwarMay28,2013at1:15AM

Ihavegonethroughyourblog.Theinformationyouhavegivenarereallyinformative.Thanksforsharingthepost.

GyminPanchkula

KantheshaMurthySBJune27,2013at7:04AM

HiKenn,Foryourconfusion

July2319915+2(2321)+1+1(19001999=>1)5+2+1+1=97=22isTuesday>youcanseeinthispageonlyontop.

RushikeshMJuly2,2013at12:26PM

Dearkanthesta,Why1isaddedtwicewhenweknowfor1900to1999itstobaddedonlyonce

RushikeshMJuly2,2013at12:28PM

Solvefor01march2011

RahulAskarSeptember23,2013at9:52AM

NIceJobPi!Youcanseemyeffortson:calendarinmind.blogspot.com

DougBangJanuary7,2014at5:48PM


DougBangJanuary7,2014at5:58PM

Itriedthiswithmybirthday:June2,1978whichcameoutto4=Thursday,butwhenIcheckeditonline,mybirthdayfellonaFriday.AmIdoingthemathwrong??

June=3the2ndofthemonth=21978=2072+6+1=6+6+1=13



Therefore,June2,1978=3+2+13=18

and18mod7=4=Thursday!

JohnJanuary10,2014at12:02PM

HiDoug,Ithinktheproblemwiththewayyousolveditisthatthereisaleapyearinbetween2072and2078whichwouldincreaseitby1.

Ididthis207856=2022.Iknowfromthechartthat2020=4.So4+2=6.Addthe1foritbeinginthe1900s.6+1=7forouryearcode.

3forJanuary,2forthedate.3+2+7=12.127=5.5isFriday.__________________

IwastakingthequizandthereisadatethatIcannotsolve.January24,2068.Thequizandotherplacesonlinetellmeit'sTuesday,butIkeepcomingupwithWednesday.January=624canbefactoredto=3206856=2012.Charttellsmethatis=1

6+3+1=10.107=3.Wednesday...IstheresomethingI'moverlooking?

GuyBurnsJanuary10,2014at11:34PM

Doug78=76not72

JohnJanuary13,2014at4:55PM

Nevermind,Ifoundmymistake.Thisparagraphmusthavegonestraightthroughmybrain..."Note:It'simportanttonotethat,inleapyears,Januaryreducesbyoneto5,andFebruaryreducesbyoneto1.Theotheryearsdon'tchangeinleapyears.Leapyearswillbediscussedinmoredetaillater."

JoshJordanJanuary14,2014at10:34PM

Iliketheideaofaquizoneachphaseofthecalculation,buttheFirstSundayDoomsdayAlgorithmissimplerandeasiertoremember.http://firstsundaydoomsday.blogspot.comThemonthscodesareespeciallyeasythere,becauseofthemnemonic"Iwork95atthe711"(SothemonthcodeforSeptemberis5andthemonthcodeforNovemberis7),andthateveryevenmonthisitsownmonthcode.ForinstancethemonthcodeofOctoberis10,orAprilis4.ThesetworulescombinedcovereverymonthexceptJanandFeb,whichalsohaverelativelysimplerules.

AndrewMcDermottFebruary17,2014at2:29PM

CaptainNegatorycommentedthatfindingthenearestleapyearwascausingtrouble.I'vefoundreducingtheyearby28,56or84firstsimplifiesthis.

Forexampletogettheyearcodefor2091I'dthinkofitas'WhatdoIneedtoaddto84togetto91"thenIsimplyuse2007asmyyearwhichis2004(5)+3=8=1.

ThiswayIcangettheyearcodeinacoupleofseconds.

Hopethishelpssomeone!

UnknownJune21,2014at6:03PM



What'swintergottodowithJanuary?Ohright,someofyoudon'tknowthesouthernhemisphereexists.

AkiDakhaneAugust13,2014at11:58PM

CananybodypleasetellmehowtocalculateJanuaryandFebruaryinleapyearbetween20002099.Example:7February2016

MaxworthMineralsAugust19,2014at2:17AM

HiAki,

Youmaytrythis!

Eg:7Feb2016

fortheyear2016,thecodeis6

forthemonthFeb,thecodeis1.Sinceitisaleapyear,youneedtosubtract1fromthemonthcodeandthisappliesforJanmonthaswellforleapyearcalculation.

Forthe7thday,itis77=0

Now,Daycode+Monthcode+Yearcode

0+1+6=7whichis0(77=0)

0standsforweekdaycode"SUNDAY"

Hopeithashelpedyou!

KUMAR

AntonioManzanaresJimnezOctober5,2014at6:39AM

Hola.Megustariasaberculessonloscdigosdeldamesyao.Gracias.

KimLucenecioOctober24,2014at8:34AM

what'stheyearcodeof2015?andwhy?

MartinOctober25,2014at11:01AM

ThemnemonicsforthemonthnumbersmakenosensetopeopleoutsidetheUnitedStates.

RobertioNovember3,2014at6:54AM


srikanthsNovember9,2014at7:23PM

thankyou

reginaldskibbaNovember10,2014at10:07AM

january1,20206+1+4=1111/7=1and4/74=thursday,whyisitwednesdayincalendar?didicomputesomethingwrong?

JeshurNovember12,2014at12:48PM



@reginaldskibba2020isaleapyear.SomonthcodeforJanuarywillbeWinter(6)minus1ie5.

JeshurNovember12,2014at12:51PM

Excellenttutorial!Iwon'tfindthose"genius's"ondiscoverymuchfascinatinganymore)Themnemonicshavesimplystuckinmyhead.Wonderfulteachingmethodology.Goingtopracticethisasoften.Thanksaton!Cheers!:)

eusypaulNovember22,2014at6:39AM

mybirthdayisDecember241991.basedontheformulatheresultis1monday.butitsTuesdaynthecalendar

DarthFranciumNovember23,2014at10:56PM

@eusypaul

Ibelieveinsteadofyearcode1,youareusingyearcode0.

9184=07closestto07is04whichisyearcode55+3=8(since7is3awayfrom4)8=1Day=2421=3Month=44+3+1=8+1(becauseofthedatebeinginthe1900's)=997=2whichisTuesday.

WhatIfindmyselfdoingofteniswhentheyearisbelow10forsomereason,Iskipastep.Insteadofdoing2004=yearcode5andthenadding3for2007,illjusttake2007anduseyearcode7insteadtoget0becauseitendsin7,whichiswhatIthinkyoudid.I'mnotsurewhymybraindoesthat,butitoccasionallychoosestoskipthatstep.probablywhatyouaredoing.hopethathelped.

ChristopherPiedadNovember27,2014at11:31PM

icomputed10161981myansweris4icheckedittherightansweris5.ivedalreadyadd1foryears1900..plshelpthanks

AdrienMarcadetDecember2,2014at4:35AM

Ido17thofJanuary1936overanoveragainandIalwaysfindSaturdaybutit'saFriday...Here'showidoit:1936isa3year:36/12=3Januaryis6,andfinally17is...well17!Therefore:3+6+17=26With+1forthe19'shundredsit's6>Saturday!!!Sowhy,whyisitaFridaythen??Thanks.

AdrienMarcadetDecember2,2014at4:50AM

Christopher:1981is3:81=12x6+9and6+9=15>1514=1andfinallyyouhavetoadd2leapyears.ThenOctoberis6.Nowweaddeverythingup:3+6+16=252521=4Andthe+1forthe19'shundreds,it's:5!!!Hopeithelped.

sasanksamudralaDecember6,2014at8:36AM

1stAugust1997

2097nearestleapyearis2096whichhasyearcodeof87=1



so2097yearcodeis2

1+6+2=9

for1900'sweneedtoadd1sonetresultis9+1=10107=3whichisWednesday

ButtheactualresultshouldbeFriday

RobertBakerDecember10,2014at7:05PM

@sasanksamudralaYouarecorrectthetensyearcodeis2andthecenturyyearcodeis1forayearcodeof3.

ThemonthcodeforAugustis1,thisiswhereIthinkyoumadeyourmistake.

Andthefirstisa1.

1,Aug,19981+1+3=5whichisFriday

swechasumedhaDecember16,2014at1:21PM

thanqsomuchforpostingsuchausefulpostlikethis:)

DzvqrJanuary14,2015at1:04PM

2038,January,5th5+6+5=16167=99=???Pleasehelp?

AnandBhagwatJanuary18,2015at7:48AM

IhavealoteasiermethodofrememberingtheYearCodes.ThatistomakeaperiodictableofYearcodes.thereisonlyoneruleforalltheyearcodes.Ihaddevelopedthis35yearsago.Anyoneinterestedshouldsendarequestonanandnb124@yahoo.com.Itisallfree.

AnandBhagwatJanuary18,2015at7:54AM

Sampleofthetable.forcompleteexplanationpleasewritetomeontheaboveemailaddress.

Yr.Cod5316420

L/YearsTable(codesforYears1,2&3are1,2&3respectively)481216202428323640444852566064687276808488929600

SPKAILASHJanuary31,2015at12:55AM

Howtocalculatetheyearcode...explainclearly...plzzzz

onepatasMarch26,2015at4:48AM

Hi,I'dliketoaskyouifIcantranslateyourpostintoCzechandpostitonmyblogwithalinkonthissite.Thankyouforanswer.

rohitpushkarApril11,2015at12:17PM



NewerPost OlderPost

iamunabletofigureoutthelogicforthisquestionwhatwasthedayoftheweekon17august1983?

AnthonyNeubertMay7,2015at9:09AM

@rohitpushkar

ClosestLeapYearto1983is1980.8056=24.2024=12/2=2+3(19831980).Sotheyearcodeis5+1for1900s+3(1714)+1(August)107=3WhichisWednesday

TomDaleyMay25,2015at4:38PM

Countmeamongthosewhofindtherecommendedprocedurefordevelopingtheyearnumbertobetoofussy,withtoomanysteps/shortcuts/mnemonicstojuggle.I'vereducedittoasimplerprocessformyownuse:

Fortheyearsofthecurrentcentury,simplytakethelasttwodigitsoftheyear,addthenumberof4'syouhaveaccumulatedonthewaytothatnumber,andthendoyourmodulo7.Example:for2050,take50,add12(becausethereare12foursin50),andyouhave62.Divideoutthe7's,andyouareleftwith6.Neat!

Asalways,youcanmodyour7'satanystageso50+12=62willgiveyouthesameyearnumberresultas1(50mod7)+5(12mod7).

purnendusenMay29,2015at8:30AM

Tellmeabouttheprocessofcalculatingyearcodes..plz

HiepNguyenJune10,2015at8:28PM

Bivitrthayvchitit..........................Mr.HipamMBngbong88lbong88

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