Mod 7 - NRICH - Millennium Mathematics Project

Pierce Geoghegan and Etienne Chan solved this problem using modulus arithmetic. This is Pierce's solution: Initially we use the fact that 3 n (mod 7) = 3 x ... Skipovernavigation NRICH Mainmenu Search accessibility contact Skipovernavigation Termsandconditions Home nrich Students primary age5-11 primarystudents secondary age11-18 secondarystudents Post16 age16+ post16 Teachers earlyyears age0-5 Earlyyears primary age5-11 Primaryteachers secondary age11+ Secondaryteachers Morelinks Topics Events NrichEvents Donate DonatetoNRICH HideMenuYoumayalsolike Purr-fection Whatisthesmallestperfectsquarethatendswiththefourdigits 9009? OldNuts Inturn4peoplethrowawaythreenutsfromapileandhidea quarteroftheremainderfinallyleavingamultipleof4nuts.How manynutswereatthestart? PrimeAP WhatcanyousayaboutthecommondifferenceofanAPwhereeverytermisprime?Mod7Age16to18ChallengeLevel Whatistheremainderwhen32001isdividedby 7? ZiHengLimandHagarElBishlawifoundapatternwhenthey raised3toapower,divideditbysevenandfoundthe remainder. 31=3 32=9 33=27 34=81 35=243 36=729 --PatternFound-- 3/7=0R3 9/7=1R2 27/7=3R6 81/7=11R4 243/7=34R5 729/7=104R1 37=2187 38=6561 39=19683 310=59049 311=177147 312=531441 2187/7=312R3 6561/7=937R2 19683/7=2811R6 59049/7=8435R4 177147/7=25306R5 531441/7=75920R1 Whenyoudividetheexponent2001bythenumber6,theanswer willbe333R3.Thethirdnumberintheremainderpatternis6, therefore32001dividedby7hasaremainderof6, assumingthatthepatternkeepsrepeatingitselfeverysix powers. Canyouprovethatthispatternwillkeeprepeatingitself? PierceGeogheganandEtienneChansolvedthisproblemusing modulusarithmetic.ThisisPierce'ssolution: Initiallyweusethefactthat3n(mod7)= 3x[3n-1(mod7)] so 31=3(mod7)=3 32=3x3(mod7)=2 33=3x2(mod7)=6 34=3x6(mod7)=4 35=3x4(mod7)=5 36=3x5(mod7)=1 Nowwehave36=1mod7anditfollowsthat3 1998=(36)333=1 333(mod7)=1 But32001=31998x33=1x 33(mod7)=6 thereforetheremainderis6when32001isdividedby 7. twitter facebook About Contactus Meettheteam Supportus Ourfunders Techhelp TheNRICHProjectaimstoenrichthemathematicalexperiencesofalllearners.Tosupportthisaim,membersofthe NRICHteamworkinawiderangeofcapacities,includingprovidingprofessionaldevelopmentforteacherswishingto embedrichmathematicaltasksintoeverydayclassroompractice. Registerforourmailinglist