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BT /StemV 20 N@JPH+cLa$-aj.20!4)bri$Pir-2bfi\%Y.^i>qlYhZDM,LWp`"B 2.942 0 Td (Arithmetic,) Tj A&Y7+!4<8_f71dKE4SFOMF[bcn=I5),>iR(K_;"a"F3O66q*;%ob9j8Dm`Qi &pd@Q9kf9tn'Y.-EE3RtJ,5qblHZuR&d+^[]1! 0.549 0 Td (=) Tj Q q ?A###Hu$Q@U1="b4&9Y=&3n6Zp8gJ?o?E668SHlc*qI+:\+c2p9F8)!V+::E=,'.IM'c5+u./uYXc?G,P\ak -12.883 -1.999 Td (The) Tj 0.712 0 Td (\242) Tj 1.333 0 Td (b) Tj /F3 1 Tf %PDF-1.1 endstream 11.96 0 0 11.96 0 0 Tm /F3 1 Tf BT "+E2IF$=n9u+> /F4 1 Tf 0.517 0 Td (um) Tj /F5 1 Tf 1.178 0 Td (If) Tj $x\equiv y\bmod m$ means $m$ divides $x-y$, $y\equiv z\bmod m$ means $m$ divides $y-z$, and. 1.812 0 Td (:) Tj 11.96 0 0 11.96 0 0 Tm 19.257 20.838 Td (in) Tj 0.627 0 Td (Euler's) Tj . @m^I;%.I5C2UPG^#Cq/-EXR'G$C'Xnq_@U:tcD1dW+1I$^QuT^gE`u4#lqYg(gVfHr]>W14YXQT)Vk[f?4>5:ViO]M]&&*eMM)d$D!L`*Ld'AjR?FVgE)FUHFb4*>ra-/1Yh Q q /F3 1 Tf j4qF[%G:>Q,aa/*jNr@W6P+'cH$[K3YIiTI4E=:eIEktRoj2(7'-.r81^Z2)g>e]a6GP-rJXAms-s$ Q q /F5 1 Tf 1.977 0 Td (F) Tj /Subtype /Type1 33.193 60.105 Td (\244) Tj ,ju=N>Ie65>G^\s.mZ$-C?tp:A)[WCa;UP-*TshsD'#r]8Sur3#p@filn,&.FYa7E5mO7;`YSt6*Y)=H@YZbTV=k/\0+0ba"WmCQ6>Akn1!/QtLRJlX;"c?7M=Pcd5 8F`[t$F`8HX3Ahp4F*2G@Eb0<5+E2IF$=n9u+>Yer04fB7\FEarZuF7_WDffQ2BkM+$+E2IF$=n9u+>bbp05t?? 189.12 504.48 m 189.525 504.48 l 189.525 518.43 l 189.12 518.43 l f /F8 1 Tf 1.120 0 Td (instance,) Tj NTQt(2S'^(hqbG;^=phJ47l(N'j)6p]Zgf!U*-l"R/)E'Qr9M=AfQpoXc3p9i`"=otcEhYN+1[\$&[QUCCp,b+sa97_M1^B%l6-_^Rp`e=/ 7.97 0 0 7.97 0 0 Tm o_2';umj,6hpKosW>Qg7.6QBmLGaaLVo^o_j3r*u\c)%^LnAnm!^>UYs,D6CSZoq;OQ%FdZmr4+:E_ /F4 1 Tf 1.812 0 Td (21) Tj [m[%-k.:s9nQC+9Z"(uj1L6o-3rm]\0enpg37aEF@,]_a]=qjMDWMGoOe;bDIAFfS/6aT.':\KPRHRdVfQ!RA1(s9j[!M`!l7! 0.959 0 Td (=) Tj 1.992 0 Td (of) Tj /F1 1 Tf V.h/ZP\e_>PR?P$I1dg$8H`'@O3Oa!eXPnZ_'2/\"L#Og0B 1.501 0 Td (\() Tj 11.96 0 0 11.96 0 0 Tm 12.965 26.508 Td (,) Tj '58bgATQA<2uiX`a,c9[9+"2O/L6H`66?uN.^.dN;b%j(on^7":l!bU7pYHC7e3K[`G4ns[\7F*r[$ !1'n0;Z03+O5d7X3u*AOM\tB:Vg:)n9ZE_8JK_T'TX*,%XEt`fcpD/G?Ru=,_/t5cn3A=[f%&2]7q? 1.113 0 Td (lik) Tj 1.648 0 Td (=) Tj 1.250 0 Td (en) Tj Q q 23.365 40.920 Td (=) Tj Tj 0.877 0 Td (100) Tj endobj Q q 2.121 0 Td (\301) Tj ]Pf[nTEhIMh4='dBX@uBNe?JSG=S68Hdi'0`rH` /Subtype /Type1 1.322 0 Td (5) Tj Assuming the two solutions are incongruent such that: x0 + (m/c)t1 = x0 + (m/c)t2 (mod m) _____ (3). 1.922 0 Td (n) Tj 0.558 0 Td (or) Tj @Wq=1F5,b^#!&dlP\gj /F9 1 Tf 170JG170JG170JG170JG170JG170JG170JG170JG170JG170JG17$=dm`@ 2.100 0 Td (are) Tj /F4 1 Tf 11.96 0 0 11.96 0 0 Tm 0.664 0 Td (in) Tj ET 189.6 490.08 m 287.754 490.08 l 287.754 490.485 l 189.6 490.485 l f 1.038 0 Td (f) Tj 1.489 0 Td (\(mo) Tj 327.36 574.56 m 422.256 574.56 l 422.256 574.965 l 327.36 574.965 l f 33.193 55.554 Td (\244) Tj Best practice is shown by discussing some properties below. $&k9T+5]@6uA=l9'\rh]o-&`(O#(DqBo@^X95"T0B*OQfM_I$_->RXBE;"^ 1.130 0 Td (a) Tj Y&3mRm]/TPh!nHD[KU>"TTVGCKSOZ/[D@;iJanp,FCI\59nPZCC>^,4EWl]InbR9&`n%*e(!d[$HC5 0.500 0 Td (\(5) Tj Thus the relation " \(\equiv\) "on \(\mathbb{Z}\) is not antisymmetric. 0.771 0 Td (F) Tj DJjim%.#_N0Ir!LZY9uce-m"0IZHaqPcb? 0.777 0 Td (1) Tj 11.96 0 0 11.96 0 0 Tm /F3 1 Tf 20.884 42.690 Td (is) Tj QV7#h8Y-I]+$9(nkrM9l;+nm]6_d\*lfPQ[q+jU&)[5'0EZ? /F8 1 Tf -27.749 -1.166 Td (\301) Tj XF:X&*8iJU0/].%,_Jb_sg)MctEbbQM(%jUfC;LPB/6/"\Sh#2t"TnYE0! /F7 1 Tf endobj 0.789 0 Td (ulti-) Tj /F5 1 Tf To solve this problem, we simply divide both sides by 9 to get that , and that is our answer. [eN8,.B>\La"KR68f[/J^@V3E1RN;_:_s,8$0cH_Me9Q@GN%0*2MSp+ You then change the equation to a congruence modulo using the smallest coefficient. 208.32 504.48 m 208.725 504.48 l 208.725 518.43 l 208.32 518.43 l f ET Tj 11.96 0 0 11.96 0 0 Tm /F11 1 Tf ET 1.295 0 Td (\257nd) Tj 23.235 56.580 Td (1) Tj Heres a trick, start by listing the equivalence classes as separate columns and then start at zero and keep writing numbers consecutively, wrapping to the next row, as shown below. 0.381 0 Td (n;) Tj F`[t$F`8H\0d&,kBQ@Zq+E2IF$=n9u+>kns04o-FEZf=DF:)Q$E$. 1.038 0 Td (8.) 1.949 0 Td (of) Tj >> 3.242 0 Td (than) Tj /F3 1 Tf !u.MriA(7[Y@i296k2Mmf(cT`Zi0HIgLcWqVK%9!518QQh]b6 /Length 92 0 R ?0oD..O#@ps0r;f?/[ATW2M+@0g[+AuodBQS;o(@:X:oCj@.6AS)9&7W30d:18!N+>6)V;DBpLA7]?[02Q(kDKJj'E+L. 1.793 0 Td (wing) Tj Tj 1.849 0 Td (e;) Tj Arithmetic modulo m Let Zm = f0;1;:::;m 1g. /F3 1 Tf To prove the transitivity property, we need to assume that 1 and 2 are true and somehow conclude that 3 is true. 0.789 0 Td (v) Tj Q q 7.97 0 0 7.97 0 0 Tm 1.039 0 Td (b) Tj ]*qHJ'8=C!pHFL)Pk+s&/Uo4T]f)?QO"kW]k Now that you know 16(20) is congruent to 1 mod 29, multiply both sides of the equation by 5 to get 100(16), a congruent to modulo 29. 0.389 0 Td (a) Tj 1.322 0 Td (0) Tj /F3 1 Tf !rb\0[-=E:d*XYsPjF?\6M""/(8cZN@ib?lOOh'[(.0MY.&2;%7VrZAj1C**Br*ITLU$_S 1.322 0 Td (1) Tj /F3 1 Tf /F4 1 Tf /F5 1 Tf 1.242 0 Td (and) Tj NqX%Bi_"hK8iYQPnX;[`kn6$X5H_GUo:Eo_f"/I=Oi7Uj:"+n6^T)*lN9:8r!>/=]c"B0b1^GeVaJm 3)t4Bl%^*+E2IF$=n9u+>Gc7+>6Q)BcqA;F:)Q$E$-ko0H`#FF(lb9F`[t$F`8HX2_uR0:MjZU@3BN 2.501 0 Td (are) Tj 1.359 0 Td (prop) Tj 11.96 0 0 11.96 0 0 Tm iPQWaQC/kR1Ln>1n:/7G.57)W[5. /Annots [ ] /F4 1 Tf 4.648 0 Td (algorithm) Tj bmIAQL9$:ofZ$FY\O4$.0eG`4L.%s'ItuUUMNQ5%Fo&C#kX*n[D]lku /F8 1 Tf 0.746 0 Td (> /ProcSet 4 0 R >> -12.569 -1.999 Td (4.) /F7 1 Tf BT /F15 1 Tf /F8 1 Tf 0.959 0 Td (my) Tj 21.743 76.911 Td (3.3.) Q q 11.96 0 0 11.96 0 0 Tm 4.073 0 Td (mo) Tj "+E2IF$=n9u+>bqu05,WMBOu'(E-670A9Di61c?I0E+O'(De'u0F`[t$F`8H[2]sbpCghBu+E2IF$ 1.030 0 Td (the) Tj 7.97 0 0 7.97 0 0 Tm /F3 1 Tf g$G]RU#(A*)o&ZcfS7VnL3C)T7Q%c+`\`8TBKrfI\$h[@C`P=J F:)Q$E$-kh2'=Pd+E2IF$=n9u+>GQ.+>7IPE-670A9Di60ebO/05D\LF`[t$F`8HX0K:1.C`m\>F:) H&oX]7("fh=p&B91fGjl*Nb50$d]'7gBN;!Sj-G:p#+dOZDh*>31+9HA5-4bI"ttMHoco7C:F04aD4 /F5 1 Tf 34.767 31.885 Td (\)) Tj 1.333 0 Td (erational) Tj 0.924 0 Td (e) Tj /F3 1 Tf /F4 1 Tf )"+>6M5E-670A9 >'.ClE_'@uY#]?YXHCF31RXT&7'="p:hde6cBn#EdJ;?Ud#F#Q[?eU]=+V7"O8u%8dujN:)i7VkieH 33.195 78.087 Td (m) Tj /Type /Font ;#&L`dBW!I$8 /F3 1 Tf 2.011 0 Td (1) Tj kp3$9knBcqA;F:)Q$E$-kp3?TtqBQ##4F`[t$F`8HX3A;R/F(o9(@3BN3F:)Q$E$-kq1E\>iBPAT.F 1.322 0 Td (4) Tj 4.464 0 Td (Z) Tj 0.707 0 Td (relativ) Tj Q q 0.746 0 Td (=) Tj 43.312 81.987 Td (\244) Tj /F3 1 Tf ET dt%6/iYr@(*MJ203t(9kCWC#kS)H+3;C=H&8enM, 4UWGN^0W%:]WQu#"fhDDRVJL\u674ZjZ18/AEKKN: /F3 1 Tf << Does a chemistry degree disqualify me from getting into the quantum computing field? /F5 1 Tf 3.189 0 Td (to) Tj 0.849 0 Td (pro) Tj 1.255 0 Td (m) Tj %PDF-1.4 )$3lh9&g?A.SE2Ni8E%8]U2>riU(fCsf\8TJ7<1gEPL6le That is \(7 \not \equiv 15(mod \, 5)\). 343.2 426.24 m 343.605 426.24 l 343.605 440.19 l 343.2 440.19 l f 27.806 34.809 Td (0) Tj cp69_p62j(mXB*Qt@ICd9]sFN;EX5Eq",g\Nre\tYlY%>E,Q%srP-G1ppn]oif1e^N[9=d+Ma#\*G?8dR998R>2?B^f54"I'T+@bnNcTrN^PEi:$Vi;s)*j93br?6Q2j/U$(ca endobj 0.982 0 Td (=) Tj >6`6F(oH/DBNn@F:)Q$E$-kn3?TtOBPAT.F`[t$F`8HX2_Z@-:ig=nE-670A9Di60fLp304nf=E-67 0.500 0 Td (1) Tj Q q /F3 1 Tf /Length1 2627 >> /F3 1 Tf 11.96 0 0 11.96 0 0 Tm 44.558 78.868 Td (m) Tj Q q /F4 1 Tf A#WaG[Qb:>)+OrQ5/h"e!3'Pa^B9PhcRYb!\Qb[g=t*e:tre's5s>@uk$$*M?SiIp711M[ao^".h&4 /Length1 2008 1.686 0 Td (follo) Tj Okay. /F4 1 Tf /F3 1 Tf /BaseFont /CMMI12 11.96 0 0 11.96 0 0 Tm [INp)ljXga5U`/sq&TTrC[I-XOIPtT?uGhqc7I?ktSW-j]3H1T!1A.$eIqdgLens>V:gpHhiH``,pV!t 2.592 0 Td (n) Tj /F4 1 Tf 23.411 18.035 Td (\264) Tj Another Example Let d = gcd( a;m ) = sa + tm . 1.039 0 Td (3) Tj 1.812 0 Td (20) Tj 11.96 0 0 11.96 0 0 Tm -26.277 -1.166 Td (that) Tj +>77JE-670A9Di60eb=)04c8FF`[t$F`8HX0JXb(Afu&8F:)Q$E$-kh1E\>`+E2IF$=n9u+>GQ,+>7 11.96 0 0 11.96 0 0 Tm 0.680 0 Td (o) Tj /F4 1 Tf 1.567 0 Td (\241) Tj /BaseEncoding /MacRomanEncoding 7N=:[\=gbVDj]icXed%MuaU`G,]RG==Zj2uT/b54ri,k7Z79X$@(@oO7&H/I4YhQ26#E,2OorkU[OG-8_(VUp 2.497 0 Td (that) Tj e!(bZjYSE\RVJPZ6Hhb%MN.#V-,Rr3j=i_&;c%6(b4``*>4TtD[8*``n1c7BM8)k_ah(M-HAK/Z^c)?Ad;Z.o;XKNfW(ri>&? Q q 11.96 0 0 11.96 0 0 Tm 32.487 36.670 Td (in) Tj Second, this is the first step in building the tools we need towards working with some . 1.281 0 Td (the) Tj -8.368 -1.999 Td (3) Tj l[bX)5u)W%B>aj)+n\ZTLjR=/\o$U;=`k>T!=B&K^qn 2.054 0 Td (m) Tj /F14 1 Tf ;2F`[t$F`8HX2`Mp5AT g9U9H;'[*/'aiU$,h&=&!YaL^d^*D7F%gZ7s1?fY-J! >b25..4"!d#! eg`.%kFJ9hhA17@)NQ&:tCfl]q%V;Y]Fs$Z;i.-n8c9BSuuhq_[t1\J9MoXB9-So&/i=-p0JBlrn"- 0.571 0 Td (er) Tj s0&?&d`O4T6*0\mmQlPP5EiZ>Zq*!dG_\HWjN$Crs8OCF]8`U#O*"$c"XM[BJq51o0N jNkJB'M#/*^5F]J+Bq_)=[9ITd0CpiKP@S#]$JKY];#6i@gY74G;;a-#7mejDV$k#lB 2.879 0 Td (class) Tj 1.038 0 Td (f) Tj 0.354 0 Td (w) Tj CNE-670A9Di60ebI-052PJF`[t$F`8HX0K(%,C*7JGQ0+>7ORE 1.129 0 Td (if) Tj 343.2 370.08 m 343.605 370.08 l 343.605 384.03 l 343.2 384.03 l f Q q '3i'q.RI(7tiiokQimeSG6Rre[Eqd+"S.cJZ1?pE:>il9/BI:5%(T'$Uh[9aT[ XP`:Xl:5I)!`7[3S^4Xr*h=72*(VAhs@8B-.H83T.S!#tai50sO*4Brku^U"(JuH+]Cm=-Jb5`0$V& 1.463 0 Td (\)) Tj 2.850 0 Td (n) Tj 1.916 0 Td (b) Tj /F11 1 Tf 1.564 0 Td (e) Tj /F3 1 Tf 287.52 490.56 m 287.925 490.56 l 287.925 504.51 l 287.52 504.51 l f -29.009 -1.166 Td (\(mo) Tj 1.000 0 Td (1\)) Tj 1.300 0 Td (\270) Tj BT ,LimMb*5Gf%=Q#^T,>(%76#Du&ot(EFht`YCT;cndVA tells us what operation we applied to and . 1.323 0 Td (11) Tj c92)dQR2$tWP6Z6phEbT0"F^/ho%6.3NYB@:X:oCj@.6AS)9&:2b5g@q?c26Z6phEbT0"FdART+fDJXS@A7]? /F5 1 Tf 7eIG%GP6E-670A9Di62)ZR1ASGdjF7VHDIjr0F`[t$F`8H\3$9kaDes?0+E2IF /F4 1 Tf :e#u)eL/WTjUbR$[=]K.00mS5*C`>ant.8dIpP*<2BTGXegDUT7q^7C\B7T3,mu# 11.96 0 0 11.96 0 0 Tm 16.460 23.228 Td (that) Tj /F4 1 Tf 0.877 0 Td (64) Tj 1.600 0 Td (or) Tj 11.96 0 0 11.96 0 0 Tm /F7 1 Tf /F3 1 Tf -27.731 -1.166 Td (to) Tj 0.947 0 Td (with) Tj /F11 1 Tf Q>^ajT`#6R("Dd$9(M=FW%kkmki&*g/9G_utddZY)e0*:l#`:HdVW/4pR4tK/-6VCXUdVK?j"dpV&@ 1C@Z-:V,l^GP:#t@1+o:05.0uS:/+2@$1]>.H"05X;#*dQ'.SmOC='WP,5%8QM:i-,87H?#C&YQN>N How to get the same protection shopping with credit card, without using a credit card? /F3 1 Tf The number m is called the modulus of the congruence. +>77JE-670A9Di60eb=)04c8FF`[t$F`8HX0JXb(Afu&8F:)Q$E$-kh1E\>`+E2IF$=n9u+>GQ,+>7 0.489 0 Td (;) Tj /F4 1 Tf lSm1tn%5qFRL6LRI81(&MXG3I&mcK6d[0[B#V]135i@,*2m+Em\Ep//BC(cS8J8K2t0`U#:T\bpD.= Notation: means that a is congruent to b mod m. m is called the modulus of the congruence; I will almost always work with positive moduli. 11.96 0 0 11.96 0 0 Tm 30.456 28.840 Td (wing) Tj 0.680 0 Td (a) Tj (The LaTeX command ncong is for the congruence symbol =in elementary geometry.) 1.243 0 Td (\301) Tj ET 2.324 0 Td (algorithm) Tj 0.711 0 Td (\242) Tj 1.323 0 Td (2) Tj /F4 1 Tf Let \(m\) \(\in\) \(\mathbb{Z_+}\). i)LSg+_0\/]p@GZY\*I&0g0B'#t@r$bbnmRWg*'LL&B9;fZQ212^HbH@$a6):S9;AE]]U#:"3V,P/?T(&WmnHZ-q[?a-JimT`qHF 1.529 0 Td (en) Tj ojnCY/Tuu`"==]bZKTHnum,sN;oa91m(&+It0lA4^CpOVp+/^ima!s^;2G*5FdTYm/FKJ.G_3;NgJB Note that the following conditions are equivalent 1. a b (mod m). 0.584 0 Td (\)) Tj #.eO0G!,n7!`1%'BIqlGOk)[qTOaFAV%'b.5Tiloceqp@+. ET S8LJ)MG&E.ba/5^b#H68F^F#5j,(WpM6QM`Ljt68[,3T"d/DcZ7dT]qlBijlLA6S_lYkD6_p/T/:VA -26.163 -1.166 Td (used) Tj Q q 11.96 0 0 11.96 0 0 Tm 22.068 49.521 Td (p) Tj Tj Kd8\fUqiP=\UR_N-(1\A(ODqmji,]O$7nHm-e5eG)e]MTY+frB#%fp/sY=p 0.706 0 Td (elemen) Tj 1.878 0 Td (denote) Tj -0.107 -0.990 Td (9) Tj DJsW-AS*'(F``$SAoD\sA9Di60fCa/01TGfD.+Q+F`[t$F`8HX2DQC.6tL1V@3BN3F:)Q$E$-kn1E\ XeHIK`l!?! /F15 1 Tf /F3 1 Tf /F11 1 Tf 0.779 0 Td (+) Tj /F3 1 Tf /F8 1 Tf /F3 1 Tf Tj 4CoJYD>\5ik(H!Wf^Tu*YGmG>=EP!\5IeiZHU]NYE@tpNNk>6Z(k1_AMQ#181b;&r@X,T1nE7C]]k/>5lsttbth)#inJUP)1N1Mehb&m0kTi,ZZNIuIh:7aSSAsGk, r7F:)Q$E$.%r+>6#'E-670A9Di62Du[266L5iF:)Q$E$.%t+>6))E-670A9Di62E2g46m-GkF:)Q$E The relation " \(\equiv\) " over \(\mathbb{Z}\) is reflexive. /F5 1 Tf An important lesson to learn here is that we can only divide by 9 because 9 and 11 are relatively prime. 3.483 0 Td (in) Tj 1.322 0 Td (4) Tj 2.868 0 Td (encryption) Tj /Parent 3 0 R /F4 1 Tf /F11 1 Tf /Encoding 31 0 R )6uGh/i:OBEhFZ([UY,.3:VY%c%C>dD([cY'l@?4=I6.mFV\)*q,;hV?`$YDPtXI,\T]4%,o@8(2 /ItalicAngle 0 3.969 0 Td (p) Tj >> ad$][G%agN7Od_AD(>'43kAi+_,Ml#61]mAY9g0=r6(,Wul0e0m8`tH?5EBO]G/Gq58u>kP 1.259 0 Td (the) Tj /F5 1 Tf :[Q^;I$N_jqiJRMWh;:l&QQ:O*t[\P!SnpS,.JmSbfjqOE3JL56B: 1.812 0 Td (:) Tj 1.631 0 Td (That) Tj 1.297 0 Td (course) Tj (e,_* 1.322 0 Td (3) Tj 23.396 22.476 Td (\264) Tj /F3 1 Tf 1.322 0 Td (4) Tj /F4 1 Tf 3.687 0 Td (:) Tj So the answer is 4! 1.087 0 Td (In) Tj 1.753 0 Td (prime) Tj /F4 1 Tf 1.322 0 Td (3) Tj 6.727 0 Td (Mo) Tj 3.282 0 Td (theorem) Tj 1.269 0 Td (+) Tj Suggested for: Congruence modulo m Solve the simultaneous congruence modulo equation. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Q q 11.96 0 0 11.96 0 0 Tm 10.592 37.994 Td (class:) Tj 7.97 0 0 7.97 0 0 Tm 1.039 0 Td (3) Tj 343.2 398.4 m 343.605 398.4 l 343.605 412.35 l 343.2 412.35 l f 438.24 426.24 m 438.645 426.24 l 438.645 440.19 l 438.24 440.19 l f 0.707 0 Td (F) Tj 0.438 0 Td (:) Tj /Widths [ 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 272 490 326 326 816 326 272 381 381 326 762 272 326 272 490 490 490 490 490 490 490 490 490 490 490 272 272 326 762 326 462 762 734 693 707 748 666 639 768 734 353 503 761 612 897 734 762 666 762 721 544 707 734 734 1006 734 326 326 272 490 272 326 326 272 490 544 435 544 435 299 490 544 272 299 517 272 816 544 490 544 517 381 386 381 544 517 707 517 517 435 490 979 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 612 816 762 680 326 734 707 762 707 762 326 326 707 571 544 544 816 816 326 299 490 490 490 490 490 326 435 490 707 762 326 326 993 762 272 490 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 ] /F4 1 Tf 11.96 0 0 11.96 0 0 Tm 1.746 0 Td (=) Tj 9FPpK=A"1,AsSPc@! 1.441 0 Td (duct) Tj 2.556 0 Td (w) Tj 0.856 0 Td (\)) Tj 2.542 0 Td (ALGORITHM) Tj 0.489 0 Td (;) Tj 1.494 0 Td (ws) Tj )Bfs*G\O]!qi? iPQWaQC/kR1Ln>1n:/7G.57)W[5. 11.96 0 0 11.96 0 0 Tm 37.628 52.186 Td (erse\)) Tj 2.166 0 Td (they) Tj Q q 4.835 0 Td (it) Tj 11.96 0 0 11.96 0 0 Tm s>)hoN`Y=FlZ-qp4qi^! /F3 1 Tf ;G)04Q,DF`[t$F`8HX0JFV&A0 0.529 0 Td (\)) Tj Tj BT /F5 1 Tf ^%NnC5gL39,9BRk*FHN;H/gkZQH@FSX;=R\2^f$].m52lmclS(WRM$n-KqBG(cfCGR9fFn?D-6- 1.006 0 Td (v) Tj G+Co%q$84keDKJj'E+L.H+Co%q$84keDKJT]FE2)?+BpNP0JG4(0H`%l0J5%50d&.m0O5fAART+fDJ 11.96 0 0 11.96 0 0 Tm 31.471 9.037 Td (b) Tj 0.556 0 Td (]) Tj 3.325 0 Td (and) Tj /F3 1 Tf 0.762 0 Td (in) Tj ojnCY/Tuu`"==]bZKTHnum,sN;oa91m(&+It0lA4^CpOVp+/^ima!s^;2G*5FdTYm/FKJ.G_3;NgJB /F8 1 Tf 0.856 0 Td (\)) Tj /F3 1 Tf 0.489 0 Td (;) Tj Definition. 1.450 0 Td (b) Tj /F3 1 Tf 9 0 obj 7.97 0 0 7.97 0 0 Tm 15.132 55.685 Td (it) Tj 1.495 0 Td (Z) Tj Q q 0.500 0 Td (4) Tj 0.513 0 Td (]) Tj 11.96 0 0 11.96 0 0 Tm 15.008 50.187 Td (en) Tj We read this as \a is congruent to b modulo (or mod) n. For example, 29 8 mod 7, and 60 0 mod 15. 1.039 0 Td (1) Tj 1.338 0 Td (h) Tj 2.687 0 Td (Algorithm) Tj /F11 1 Tf 1.322 0 Td (0) Tj 33.632 74.787 Td (n) Tj /CapHeight 705 0.789 0 Td (v) Tj $=n9u+>l/%06:]ABk)'tDf-\:F`[t$F`8H]0d&,\EHP]++E2IF$=n9u+>u"u06),GF*),7DBNn@F:) 26.918 54.223 Td (!) 0.737 0 Td (suc) Tj 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm 19.302 45.254 Td (;) Tj 7.97 0 0 7.97 0 0 Tm /F4 1 Tf )'OGih3W6GHN#)O(3-UO%E#0h#U?h]Ba=S?_3\#p 0.438 0 Td (:) Tj Tj 2.808 0 Td (b) Tj /F5 1 Tf c^qGC!7W'W/`Ea4N0Uk#\sMgWGnrg*t$,Uhkl6Z`iYL-^ /F4 1 Tf 0.762 0 Td (in) Tj 1.686 0 Td (message.) .pmZ>eVk[OXT^(ZbAm9N6Icmdl:nDMKY`iHS'rd2M?k!-eEBEpZHaoc'h*)UL2s21/`s"$6 22.793 42.672 Td (=) Tj << /F10 1 Tf 11.96 0 0 11.96 0 0 Tm 29.470 55.685 Td (de\257ned:) Tj 1.940 0 Td (greater) Tj We start at 0 and go through 8 numbers in a clockwise sequence 1, 2, 3, 0, 1, 2, 3, 0. 2.407 0 Td (algorithm) Tj 1.005 0 Td (a) Tj /F5 1 Tf 8.97 0 0 8.97 0 0 Tm 3.154 0 Td (t) Tj 327.36 546.72 m 327.765 546.72 l 327.765 560.67 l 327.36 560.67 l f ? [01BbgDe*R"B-8re2BZFKEaa5#0H`(m1,Up3HSHXWDId='+D#e-BHSr\ 18.652 34.689 Td (p) Tj 1.178 0 Td (All) Tj 0.517 0 Td (um) Tj c5XPo_eq4bV(*hBfmG$"g:3R)ADq4&%);S3@q*mZga4!R7nZPW-^F3Gg"_#1gu5H,ghe=hL;5?lot3s3$qbMW:PZL=DQH\nMr`3&sR]_h. /Type /Page 1.038 0 Td (f) Tj 1.686 0 Td (Euclidean) Tj /F4 1 Tf 7_Gj7E5V[[GZrhZno_tU55#fE,9tO)G&&Gl.J?#kK,VR`>MZ^1d3=]"[HD3p)A*HeYQ$Z=JNA"+V2p 11.96 0 0 11.96 0 0 Tm 'E]UYi This tool also comes with detailed learn sections and step-by-step solutions! lT6[+1FS6%0C$K&n27AL8r3:FhC#a"O_s)g8r=2j?ih98gs$'=m3Ji@dFa\E@IFb]IY65T`,t#I%p+?%qZAuB`q?Q(If\dfa!.k54Q-A,KWUum5bW=9&=/&l 0.856 0 Td (j) Tj 1.610 0 Td (p) Tj Tj (>f.JG4NM; << 0.856 0 Td (\),) Tj 0.877 0 Td (64\),) Tj 287.52 412.32 m 287.925 412.32 l 287.925 426.27 l 287.52 426.27 l f [>-[Sdu-<=k5j(NE_K+ /F3 1 Tf 8.97 0 0 8.97 0 0 Tm . /F3 1 Tf Tj i)LSg+_0\/]p@GZY\*I&0g0B'#t@r$bbnmRWg*'LL&B9;fZQ212^HbH@$a6):S9;AE]]U#:"3V,P/?T(&WmnHZ-q[?a-JimT`qHF 1.322 0 Td (4) Tj /F4 1 Tf 2.306 0 Td (\301) Tj 0 -0.990 Td (7) Tj t$F`8HX0f'q*E?K4CF:)Q$E$-ki1a"Gl+E2IF$=n9u+>GT.+>7dYE-670A9Di60ekR/06A=UF`[t$F /F3 1 Tf 1.601 0 Td (follo) Tj 1.923 0 Td (the) Tj ASu7\MF`S[EF`[t$F`8H[1a"G]Df' >> /F3 1 Tf 11.96 0 0 11.96 0 0 Tm (The LaTeX command ncong is for the congruence symbol =in elementary geometry.) 26 0 obj 22.793 38.124 Td (=) Tj /F11 1 Tf Q q /F3 1 Tf Tj /F3 1 Tf /Pages 3 0 R 1.616 0 Td (giv) Tj Q q NqX%Bi_"hK8iYQPnX;[`kn6$X5H_GUo:Eo_f"/I=Oi7Uj:"+n6^T)*lN9:8r!>/=]c"B0b1^GeVaJm ET 0.517 0 Td (y) Tj 11.96 0 0 11.96 0 0 Tm 21.453 38.124 Td (Z) Tj The railway time starts at 00:00 and continue. 27.806 42.545 Td (3) Tj /F3 1 Tf 'H(,YG.6Q!ZM_;>6T<2E>`?7G$""M'b[M0OTniZ?o1o&=%J2Ufb endobj Q q 0.698 0 Td (able) Tj 7.97 0 0 7.97 0 0 Tm 1.038 0 Td (13) Tj -3.367 -1.516 Td (Z) Tj /Type /Encoding 42.297 47.623 Td (2) Tj 2.035 0 Td (encryption) Tj Tj 1.954 0 Td (p) Tj 0.837 0 Td (is) Tj 1.701 0 Td (prime) Tj /F3 1 Tf /F3 1 Tf A reasonable number of covariates after variable selection in a regression model, Interactively create route that snaps to route layer in QGIS. /Contents 7 0 R *R0d:A_gf)q[fIPior 0.888 0 Td (some) Tj << 204.96 408 m 407.082 408 l 407.082 408.405 l 204.96 408.405 l f 0.438 0 Td (:) Tj 1.039 0 Td (4) Tj 1.332 0 Td (h) Tj BT << 1.323 0 Td (10) Tj )t`-cT]\j:4?LkpjF"on$9FK#XDk)4>DT[S'cA"p,I@Y[H:'-:PDj;EXZLR*A&=@dDH\/!7=G\@]HB 0.571 0 Td (ositiv) Tj @Vaf(`S8ndC"cI:1f?aYT6-0%uDBEh6R-8@ql$H'AaEC=q09O+B:XI#APT' BT 0.438 0 Td (7) Tj 0.984 0 Td (denoted) Tj /F4 1 Tf Q q 7.97 0 0 7.97 0 0 Tm [02Q(kDKJj'E+L.G+Co%q$84keDKJj'E+L.H+Co%q$84keDKJT]FE2)?+BpNP0JG4(0H`%l0J5%50d 0.438 0 Td (:) Tj =5*OYhB`b$]"KaLKgRs(RW8\Xl;,A,rtjekeU]8X'eT\:5f!jo))a*hq:gCF!$.Wo(p1pHlNYIM[i, 0.517 0 Td (umerically) Tj 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm 23.310 50.770 Td (wing) Tj /F8 1 Tf 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm 30.936 14.405 Td (di\256eren) Tj 0.877 0 Td (pq) Tj /F11 1 Tf ,5V9M0V"@cmC%i76=gh*Vs8p&m:Yb*W/rkG`^9;n/@J5t*Wh3sfI)_T[DQD'Le;u5p?MDQ)RLHgVJWWaET4hC%H.="85d)pc,CSY The expression -8 10 mod 9 is pronounce 1.315 0 Td (is) Tj 0.862 0 Td (t) Tj 12.091 55.685 Td (In) Tj In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder. The three incongruent solutions are given by: /F3 1 Tf UlstGe21%M4FS>^^_KS-9LF2BWI/(gJLPO.df[\)j^t$r=`)_g(R?0/Zas%G?JqRSOerqGjJcGRh)l/>B$SQ`(C[Fd>>E_[f61RcJicH(9K"N^Dg33q/]prk 8.97 0 0 8.97 0 0 Tm X_2;TfSX49%0*2hD[a@R\5S*#D!LDD>h#bVqG0RPl7D/EbdD2"(T6*VC*kd/V]`" If m = 2, for instance, these definitions say that x y, y z, and x z are even. *W>q6Y\V]%AP^jFb6P]^'[>JFDU:51JA`1ODnHnoPijKGo0L0^QXQXFn^3W3 /F3 1 Tf 1.190 0 Td (49) Tj /F8 1 Tf /F8 1 Tf /Type /Page BT 327.36 490.08 m 422.256 490.08 l 422.256 490.485 l 327.36 490.485 l f 7.97 0 0 7.97 0 0 Tm 11.96 0 0 11.96 0 0 Tm 13.580 10.203 Td (\() Tj 7.636 0 Td (RSA) Tj 1.051 0 Td (an) Tj )cG)0c$F+s]tR\fm=RV372#LKIXUKAF3%'P_h+=@/F.`T.+8jp,Vo0q$Ls9^)%SFC9;<6%lO=P62] fJiO4;I(elPMHB`>aH.J@? /F3 1 Tf /F8 1 Tf Q q 3hS)VK#g,eND0U:*XP6-I[9X?e_BXF-X&u#TRuSCW$1?pV,Vf@!LSgE.X8R1`lD/t&&ZLkMD)s:uBf /F4 1 Tf [01BbgDe*R"B-8re2BZFKEaa5#0H`(m1,Up3HSHXWDId='+D#e-BHSr\DJsW-AS*'(F``$S 1.522 0 Td (wn) Tj ff,F)u(=E-670A9Di60fUm105#<6G%De8F`[t$F`8HX3&2U0@:OG%AKYr7F:)Q$E$-kp1E\>\@Gi4+>72EAThW-E-670A9Di60fV*704SR(Bl%To+E2IF$=n9u+>Gi8+>7A=Ec#6&A7B 1.716 0 Td (secret) Tj more 4 Answers The volume of a spherical body is decreased by 10-3% when it is subjected t. more 2 Answers Young's modulus of steel is 1910^10N/m. /F3 1 Tf /F3 1 Tf /F4 1 Tf !b]#7RiH[3'!`E0[/S^*,%(2 2.153 0 Td (and) Tj /F3 1 Tf /F4 1 Tf 0.712 0 Td (\242) Tj 1.322 0 Td (1) Tj 1.849 0 Td (endix) Tj Example 2 : Find -5 (mod 3). 11.96 0 0 11.96 0 0 Tm BT 0.816 0 Td (and) Tj /F3 1 Tf 0.948 0 Td (6) Tj 11.96 0 0 11.96 0 0 Tm /F3 1 Tf 49.715 62.100 Td (s) Tj 1.322 0 Td (0) Tj Should a bank be able to shorten your password without your approval? U[KJ"7F(?_hP9>?2J`W\79n$KHZq">PnY9Z>A4J#YkO9Y&4jIl@EK9>Hl)(Hj/>5mSp'B>YGLhb+\WU If you realize the multiplicative inverse of 5 modulo 7 is 3, because 531(mod7 . \q4=$5Erg8X,jH7^gY')EiI(C%)b3gWJVsmFR[7BY-YcrDV%cME>b19b)D9:Lak6Jk,N^#N>-41`=1; Q q 7.636 0 Td (RSA) Tj mW^S"S;#WPqO$6`F=1rDU^X/A@#"0C9j_)E0D )'OGih3W6GHN#)O(3-UO%E#0h#U?h]Ba=S?_3\#p /F11 1 Tf 2.909 0 Td (ts) Tj 0.568 0 Td (d) Tj /F3 1 Tf /F5 1 Tf -0.418 -1.662 Td (a) Tj /Length1 2627 /F3 1 Tf /F4 1 Tf 1.007 0 Td (a) Tj BW#?A.3>#:P>@#(FAf=AkQM-kk6Hp>8N(%LtBK7Kmm8?J+9Cm? 1.639 0 Td (Ho) Tj qbOWQ`$bcodHM6AKXe2d0.2*?JGPCj!Z&=#qBkAWb&%._Af%_WUn*k6XJ /F4 1 Tf 1.182 0 Td (equiv) Tj /StemH 20 5.219 0 Td (the) Tj /F5 1 Tf /F3 1 Tf 1.196 0 Td (17) Tj 11.96 0 0 11.96 0 0 Tm 18.981 19.534 Td (13) Tj 5.909 0 Td (ARITHMETIC,) Tj /Annots [ ] Okay, so the remainders of 0, 1, or 2 comprise the equivalence classes for mod 3, which we write as [0], [1], and [2]. 1.206 0 Td (zer) Tj /F3 1 Tf Tj 11.96 0 0 11.96 0 0 Tm 1.000 0 Td (1) Tj Q q /F5 1 Tf 1.876 0 Td (3.3.2) Tj 1kgSb4^u$5u,',5/^2h1p<3K$A@EfFHGn`PD2IMRR6h2e@ZK\tP;l(9?l>^dhqV?=J"o`WmNjIo45' =n9u+>c%2MD]j"AF:)Q$E$-tt+>7YNAKYr7F:)Q$E$."l+>7h\D]j"AF:)Q$E$. 0.272 0 Td (y) Tj 11.96 0 0 11.96 0 0 Tm 3.536 0 Td (Little) Tj 3.105 0 Td (m) Tj 1.087 0 Td (then) Tj /F3 1 Tf ? 1.323 0 Td (2) Tj 0.870 0 Td (\270) Tj 2.067 0 Td (obtained) Tj ]JD9i'Ta40O9XpZW=uRC/$/Bn'i=S_>na /F3 1 Tf /F4 1 Tf /F3 1 Tf Two geometric figures are said to be congruent, or to be in the relation of congruence, if it is possible to superpose one of them on the other so that they coincide throughout. /Differences [ 32 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /period /comma 61 /slash 63 /.notdef /.notdef 73 /.notdef /.notdef 89 /.notdef 91 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 123 /.notdef /.notdef /.notdef /.notdef 128 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Gamma /Delta /Theta /Lambda /.notdef /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef 174 /alpha /beta 177 /delta /epsilon1 180 /eta /theta /iota /kappa /lambda 187 /xi /pi /rho /sigma /.notdef /.notdef /phi /chi /psi /.notdef 199 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 216 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 241 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] 0.413 0 Td (m) Tj 0.571 0 Td (ositiv) Tj /F4 1 Tf 2.833 0 Td (for) Tj Tj 1.661 0 Td (dulo) Tj /F9 1 Tf /F1 1 Tf 40.850 37.854 Td (\)) Tj pF6HGt*a*G)DH&DPQQVWRFDLjmEq\&B-U"EZpD(FrO*FdV%W'EZPLDtOXr.\& By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Q q 1.357 0 Td (q) Tj 1.184 0 Td (then) Tj BT /FirstChar 0 1.128 0 Td (Z) Tj /F4 1 Tf #NC)/m]4Bn2/Nan^Mk6!J%48>`@o! 1.849 0 Td (a;) Tj 4.084 0 Td (tables) Tj 0.999 0 Td (1) Tj 0.801 0 Td (large) Tj /F3 1 Tf 7.97 0 0 7.97 0 0 Tm >> j/\ujH&*Ij9`@;TGK4*W$NnRC2Prj@'m0@iA[BD&@^eDaCNOW4455JXirU5tG! (SORoEjb74`FBTftAgedE]l64c;JjO4Oe2O]5(i'-&]8fss@LpXX+rQIh[8R#o+V,_H8&3F-QTF /F4 1 Tf Modulo in Mathematics. 7.636 0 Td (RSA) Tj /F5 1 Tf stream Q q 0.571 0 Td (et) Tj /Length2 18039 2.288 0 Td (e,) Tj For example: 6 2 (mod 4), -1 9 (mod 5), 1100 2 (mod 9), and the square of any odd number is 1 modulo 8. 0.500 0 Td (1) Tj /F3 1 Tf 204.96 393.6 m 407.082 393.6 l 407.082 394.005 l 204.96 394.005 l f 0.736 0 Td (n) Tj 7.97 0 0 7.97 0 0 Tm 0.712 0 Td (\242) Tj 00!7F%MqrjH/V5sgZ>YlUK1Fg1T+d_r4'B,o,8:7^;48ELN 0.438 0 Td (14,) Tj 1.713 0 Td (t) Tj /F4 1 Tf 11.96 0 0 11.96 0 0 Tm It follows that $m$ divides the number $(x-y) + (y-z)$. ET 1.635 0 Td (\242) Tj 1.363 0 Td (The) Tj 0.924 0 Td (er) Tj /F3 1 Tf /Ascend 465 6 mins. Get access to all the courses and over 450 HD videos with your subscription. 2.255 0 Td (ro) Tj 31.602 10.835 Td (\(mo) Tj 0.438 0 Td (:) Tj Start by choosing the initial number (before performing the modulo operation *+Wsl*/m?rNlJW$9Me_I'PKfALbMlILr\>fS.ccd2MuQF3KhZ 1.739 0 Td (Giv) Tj 1.014 0 Td (49.) /BaseFont /CMBX12 7.636 0 Td (RSA) Tj >B,_$RE='p&4_*pTmghnF)4?H`lt8]:7Z;Spi8-XE$Pt%;YkV1kDUnJln7XFmU=8 2.937 0 Td (classes) Tj (SORoEjb74`FBTftAgedE]l64c;JjO4Oe2O]5(i'-&]8fss@LpXX+rQIh[8R#o+V,_H8&3F-QTF /FontDescriptor 32 0 R (T11(2k0VUbW&qo&P).77K92,YLr;Q68]VG5K&+JEKQ(/K)^=a4m+*(l0 .6AS)9&=(Q)YBQP@F6>p[N.3NYB@:X:oCj@.6AS)9&8T\BWBk'GHB5D-%0Han;AdU2*F%0kgARnVOF 2.345 0 Td (zero) Tj ]hW:9(RpP/?Nm8\/(?.\kaA?oOFmpJ>+< BT *kYc',a.L6ptqE88&6:2-fm'o5T^rJqWq/;h:7,oG<>)Hb[>USHO]t.r2;]K-2HN=sd$2p123t&Xs>`+D%Dn*HrS-d?I1,8 plGLq@gH\gLgXV(g$'`Ilh)4'gE8ogr&kA9E#Qb!ThU]^r[5C$&[PuGN\dWuVug/?+!Kf1`@E`+Oc9 /F3 1 Tf Qf^?s=T?b([,l:0m/g#7"PdM+V*I6%Jpc'0<4nKO!1]5(jY63=s`L.'$?oreAq_C7TmK-+TKB51_RBf6D%"/?;rEQLU?/Ga54dk7WK.>bL6qPa`n'Y%n0/5EMe=c@0`O"! -8.078 -1.516 Td (Z) Tj 1.038 0 Td (3) Tj 32.762 52.749 Td (s) Tj aX. 58.610 25.150 Td (6) Tj 53.946 62.100 Td (s) Tj 3.483 0 Td (phi) Tj 2.794 0 Td (pr) Tj 33.192 53.280 Td (\244) Tj /F2 1 Tf Q q Theorem 1.1. 11.96 0 0 11.96 0 0 Tm ET 2.312 0 Td (Mo) Tj ET )Q$E$-kp0H`#ZDfff,F)u(=E-670A9Di60fUm105#<6G%De8F`[t$F`8HX3&2U0@:OG%AKYr7F:)Q$ 7.97 0 0 7.97 0 0 Tm A congruence of the form [9,cF28r>+i"&mgWtSQ'4ZCaAT=PT>[*h"Jl$PQJJ1 0.847 0 Td (Otherwise,) Tj Q q *qT\dDOdlI;)I[=]HP$l 1.333 0 Td (are) Tj Q q << 1.686 0 Td (future) Tj ET /F8 1 Tf -3.758 -1.671 Td (\() Tj BT JTkTQZRMZ;3,`-?$fqeOSWE8P:\)0-hMFViFqqt9m%SB%LdafeS-/k69luf0L6(mgle=FMXq-1._bp 0.791 0 Td (=) Tj p&YF'i5C;M(##F')5#E^14VZNUk.IL/*\s>Z/+f,?bNdD:W HH. 0.381 0 Td (a) Tj `8[fV7mGk@9tO\(BH--M::n?\0g64>-[MMmFAgk`%/YE2kIVbKtJ)j? qj4aqF]W8QtoFoHq`ej/u6>K)c*H=Q983`^Sfh@@r/+#XH'>9g?ar[ZUULWLWp 1.323 0 Td (12) Tj 53.684 25.150 Td (7) Tj endobj 1.142 0 Td (the) Tj 0.517 0 Td (um) Tj /F4 1 Tf 7.97 0 0 7.97 0 0 Tm 0.984 0 Td (4) Tj 0.768 0 Td (=) Tj 0.585 0 Td (Find) Tj (W"h:g&]3-rFOtZ>tqohV8h6fFW\QS'c$YC>8)AopQ*4_IlPt\TH)W_F6bm/=q+WFBuNfStob==fFK /F3 1 Tf -^g35s! 31 0 obj Lastly, verify that 16(13)-5 will leave a zero remainder when you divide it by 29. 3hS)VK#g,eND0U:*XP6-I[9X?e_BXF-X&u#TRuSCW$1?pV,Vf@!LSgE.X8R1`lD/t&&ZLkMD)s:uBf /F3 1 Tf BT Modular Arithmetic Multiplication Addition. 0@9pR?OA>>OXTX@E.e]XutcXJpiV4/O0pZmV1G.pe)P9(6U7^Mh(;l@j*3ErogA4E!_e.uj`(S7p[F Tj M7,!X0Rmgaq]CNSOOOnOq[`6)#gFI(3)A0mK"r%:4m0tZoLm`AeWcHL:1,9$h/8? 204.96 436.32 m 407.082 436.32 l 407.082 436.725 l 204.96 436.725 l f /Type /FontDescriptor 11.96 0 0 11.96 0 0 Tm 422.4 546.72 m 422.805 546.72 l 422.805 560.67 l 422.4 560.67 l f Well, -97 divided by 11 equals -8 remainder -9. 0.711 0 Td (\242) Tj /Length2 10935 /F3 1 Tf /F8 1 Tf kM#^7`Z4H1b8R2@m69^! 11.96 0 0 11.96 0 0 Tm 36.555 15.572 Td (de\257nition) Tj The most commonly used methods arethe Euclidean Algorithm Method and the Euler's Method. /F4 1 Tf /F3 1 Tf 11.96 0 0 11.96 0 0 Tm 22.793 36.608 Td (=) Tj /F3 1 Tf B+3f72?g&]tb+=qU?f7fO4adTaK`"8gQc4@a7VE;rl/mE&BR+m-g53Y%"7m^i`C\JdS07Y:Da)%?7X :bG54lM7X7o?K\ugn=@81nKR826AW\HZdgi/T"d2>86V710_kaD )`/OGl:2BYV7C`kJl+>> /Length3 532 2.057 0 Td (but) Tj 1.749 0 Td (equation) Tj 1.038 0 Td (m) Tj Q q 450+ Math Lessons written by Math Professors and Teachers, 1200+ Articles Written by Math Educators and Enthusiasts, Simplifying and Teaching Math for Over 23 Years, Email Address 0.960 0 Td (2) Tj BT 60fLs404no@E-670A9Di60fM!504o#CE-670A9Di60fM$604ng1+E2IF$=n9u+>Gf7+>7>=C`m\>F: /F3 1 Tf endobj 1.039 0 Td (2) Tj 4.621 0 Td (e) Tj 2.452 0 Td (in) Tj We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Tj 28 0 obj The relation " \(\equiv\) " over \(\mathbb{Z}\) is symmetric. 24.390 35.676 Td (1) Tj /F4 1 Tf /F3 1 Tf tbUA?8rb^@'7F_L1PPg.o0=m-J((bqBNVR9[@k_r 3.186 0 Td (Application) Tj 0.438 0 Td (3) Tj /F3 1 Tf Tj 0.745 0 Td (\242) Tj 0.381 0 Td (n) Tj 0.556 0 Td (]) Tj (T11(2k0VUbW&qo&P).77K92,YLr;Q68]VG5K&+JEKQ(/K)^=a4m+*(l0 0.438 0 Td (11) Tj 173.76 426.24 m 174.165 426.24 l 174.165 440.19 l 173.76 440.19 l f 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm 5.449 0 Td (of) Tj /FontFile 33 0 R 3.793 0 Td (to) Tj 1.038 0 Td (8) Tj 4.452 0 Td (b) Tj `mu J<5d^hGE5:aJ--meaY6,fDGc;$-r%'H,M5B0pcd$7XIf5*F5d/(t'B(a5>B>#X8nEHd48. 1.713 0 Td (d) Tj /F5 1 Tf 0.962 0 Td (in) Tj Linear Congruence. /Length 86 0 R 22.793 35.092 Td (=) Tj ET BT 34.367 73.862 Td (6) Tj 0.272 0 Td (x) Tj 0.513 0 Td (]) Tj 2.473 0 Td (non-zero) Tj /Type /Outlines 2.909 0 Td (t) Tj /F4 1 Tf ::m>[q[mlKC1PCeC$/F(7;(!qj0R<2"^W* uf^NEMm/FMYOc9)p2(qb]mW03+d4r*UO/$XDX8SM+m1#7Vh3*E%?C5>:kkET>CHnuO5]&ClCh8;!G* This remainder (r) results from dividing the smaller of aand binto the larger. /F3 1 Tf << /F11 1 Tf GKpi0.ME^Q02TY#J?IXSr8bK&uq7%?L-QD$tsQs_ 3.171 0 Td (b) Tj A#WaG[Qb:>)+OrQ5/h"e!3'Pa^B9PhcRYb!\Qb[g=t*e:tre's5s>@uk$$*M?SiIp711M[ao^".h&4 24.077 31.885 Td (\)) Tj F:)Q$E$-kh2'=Pd+E2IF$=n9u+>GQ.+>7IPE-670A9Di60ebO/05D\LF`[t$F`8HX0K:1.C`m\>F:) 11.96 0 0 11.96 0 0 Tm 13.758 52.519 Td (The) Tj RXr0ekHhRFBBB$12at=`I_5)G?V)\9`O,,l2*)lt7CgUp_F:X7(LVB_>@"-I3F#Ml%D35':5>$&%4J /F3 1 Tf M_GdW@peQ(/Lc\O,AgZP!+aA2\7kl"NYqI[9c(t&HK2* The relation " \(\equiv\) " over \(\mathbb{Z}\) is an equivalence relation. /F3 1 Tf /F11 1 Tf /F6 1 Tf Q q Is the relation " \(\equiv\) " over \(\mathbb{Z}\) antisymmetric? 4.958 0 Td (k) Tj DWW)XICB5Q:E(1g7*XF! 7.97 0 0 7.97 0 0 Tm 5.98 0 0 5.98 0 0 Tm /F3 1 Tf /F3 1 Tf D`VADVe)4h@JHE0-uJZlYUgYf'@oDjEUqe/Ibb3IRVhd2eNA6f0erAG'k.+9*. /F3 1 Tf 1.713 0 Td (d) Tj /F4 1 Tf 2.909 0 Td (ts) Tj 22.793 39.640 Td (=) Tj 0.584 0 Td (,) Tj so it is in the equivalence class for 1, /F3 1 Tf Asking for help, clarification, or responding to other answers. 30.388 69.344 Td (9734888) Tj 2.339 0 Td (conditions) Tj 1.169 0 Td (p) Tj 0.438 0 Td (7) Tj /F8 1 Tf 4.676 0 Td (A.1.) 11.96 0 0 11.96 0 0 Tm /F8 1 Tf /F4 1 Tf /F4 1 Tf 7.97 0 0 7.97 0 0 Tm ET 17 0 obj 0.789 0 Td (ultiplicativ) Tj BT 0.658 0 Td (=) Tj 173.76 426.24 m 287.727 426.24 l 287.727 426.645 l 173.76 426.645 l f )[=d+%_,lIZTgSZI\.bD&riKdc5f7n&pW(Qgh[f4bS$KQ`]erB`+s^5>tu $-ki1E\>j+E2IF$=n9u+>GT-+>7aXE-670A9Di60ekO.0687TF`[t$F`8HX0fC.-FGT1+>7m\E-670A9Di60ek[206\OXF`[t$F`8HX1,'h(GT^sJF:)Q$E$-kj0d 287.52 504.48 m 287.925 504.48 l 287.925 518.43 l 287.52 518.43 l f Q 0.630 0 Td (in) Tj 1.119 0 Td (is) Tj 11.96 0 0 11.96 0 0 Tm 10.592 52.186 Td (gcd\() Tj 0.803 0 Td (and) Tj 1.322 0 Td (4) Tj Q q @J="0?c8\WLT7`rIPhEFud%'DX>V8qLa%(5dK(>V^oW62oKWM5I[!>+qX)iXgiMJO^^CBGEf_bYBV- /F4 1 Tf 1.341 0 Td (=) Tj /F7 1 Tf 327.36 518.4 m 327.765 518.4 l 327.765 532.35 l 327.36 532.35 l f /F4 1 Tf 0.571 0 Td (er) Tj 1.713 0 Td (d) Tj 0.762 0 Td (the) Tj :r8Df"qCB>/3I1e2\RFP.=\YtP]Z6(0>ma,UBhXn0'P[_$bM7_.g-l9U9_am/C65LI1aMMVXk,StI0>?_hrYp`d@d:Lc"RSqm 11.96 0 0 11.96 0 0 Tm 34.797 12.535 Td (to) Tj Q q Tj 1.148 0 Td (=) Tj For example, If a= 5 and m= 7, then x= 5 (mod 7) if and only if x= 5+ 7k for integer k. k= 0 gives x= 5, of course, and k= -1 gives x= -2. 0.842 0 Td (the) Tj 0.856 0 Td (\),) Tj /F3 1 Tf /F4 1 Tf Defintion: a " b ( mod m) if and only if m | (a - b). /Descent -225 T"ihNDau@5]7gdcC@>?rZGSXkS.sY$6;M,;'GEI6(6DA:De=L L`h=Zn]k@HopiQMkLEA?C%TC,[OHhEdf8OALfCAXkf91:"WMiB-jR$Ls,AX(PA$MHId[Z6eV:'4?eJ 11.96 0 0 11.96 0 0 Tm 30.437 7.870 Td (then) Tj 1.640 0 Td (elemen) Tj Every time you think about time, you use modular arithmetic because it deals with cycles of integers and remainders just like a clock. /F8 1 Tf ?1\WOrcEm2aA?tJqB5"X\Cdu_J/3TX8Yfe&"BQ6-03;JDuZfAGYO"sQqDX 0.952 0 Td (m) Tj /F3 1 Tf s&K.Ug,l9tZig,ZRTLU%_-_B. 2.355 0 Td (gcd) Tj 1.100 0 Td (17) Tj /F4 1 Tf 25 0 obj 1.322 0 Td (2) Tj 0.632 0 Td (\(100\)) Tj Q q /F4 1 Tf T_#uSdtU!=m"#e+_*H(;DfM-f_rUM/cPKo3@!7r /F14 1 Tf Our math tutors are available24x7to help you with exams and homework. The two methods allow us to extend modular arithmetic beyondsimple addition, multiplication, and subtraction to give room for division. -19.260 -1.461 Td (and) Tj 0.489 0 Td (erify) Tj 0.489 0 Td (;) Tj /F4 1 Tf ET U^'20P^hC%4@l6R8kDO.0B7.p.S2h`BG%)C,OHlE/Wf/rUFH.^^/m. 1.206 0 Td (is) Tj /F3 1 Tf Find \(18:00\), that is find \(18 (mod \, 12)\). 15.970 36.688 Td (\)) Tj ?A###Hu$Q@U1="b4&9Y=&3n6Zp8gJ?o?E668SHlc*qI+:\+c2p9F8)!V+::E=,'.IM'c5+u./uYXc?G,P\ak 11.96 0 0 11.96 0 0 Tm ET /F3 1 Tf 11.96 0 0 11.96 0 0 Tm 26.963 10.203 Td (\241) Tj 1.332 0 Td (h) Tj /F3 1 Tf 7.97 0 0 7.97 0 0 Tm 3.906 0 Td (elemen) Tj BT 11.96 0 0 11.96 0 0 Tm /Flags 4 /F3 1 Tf Tj /F15 1 Tf )dId2j!N(6,+8IMbEk?^qb.$s>[Im9RWn*^h&a60u(-Yi3IOcX?gV(Rn28E#;qleKZ7/ 1.178 0 Td (Generate) Tj /F4 1 Tf 1.039 0 Td (1) Tj '2bVa&jr 1.033 0 Td (then) Tj gE#!i-79a0Q`V+>O#4V!e`LEJQVgq5h5IdOsEDk);UVPB-8*XMp^1K2>!r:iqA5f'Mfu 11.96 0 0 11.96 0 0 Tm %]`J<9dE\_^r_5j]mIR.AaS5VGSufM9UR5`_iW=UU*h>Od,m7;oWbqJtW2Ca@g;gWA@1eT;Sn3Ol /FontFile 33 0 R 7.97 0 0 7.97 0 0 Tm 0.792 0 Td (computed) Tj 0.979 0 Td (.) ;G)04Q,DF`[t$F`8HX0JFV&A0 2.447 0 Td (In) Tj Substrat the last from the first to get x z = ( k n) m which means that x z ( mod m). 0.877 0 Td (100) Tj BT << 3F:)Q$E$-ko1a"G`Afu&8F:)Q$E$-ko2'=PaBcqA;F:)Q$E$-ko2BXYbC`m\>F:)Q$E$-ko2]sbcAn 21.743 76.911 Td (3.3.) 0.789 0 Td (to) Tj To better understand how to use the Euclidean Algorithm Method, we will use the equation 11x =1 mod 23. /Descent -225 327.36 490.56 m 327.765 490.56 l 327.765 504.51 l 327.36 504.51 l f 1.832 0 Td (r) Tj nWD5]O]<3d/C=!kCpMgcHq)fq0! 2.236 0 Td (has) Tj /F6 1 Tf 18.593 7.870 Td (mo) Tj /F3 1 Tf 0.711 0 Td (+) Tj For example, we have the following results. ]Qh9#tNY+OD`0d_G:9EHQDt+OUEstSEC*gM`Wi>0tCnDJ,-\V@Q`&`oGX*#cfE_0,l(#$h&&)*gE!6 16.399 43.711 Td (2) Tj Recall that the Division Algorithm states that if any integer a is divided by a positive integer n, then the remainder r is always between 0 and n - 1. P>(Pp0:iYuk$u)R8eQipFt'iNKu7,@;7[AaFK@tjpSa?XYH9^uY )98fMHI:mpgN1G*:E$Y$`"p"id6hP$c83d]Z_4dXmhu2V^;+. )s 438.24 356.16 m 438.645 356.16 l 438.645 370.11 l 438.24 370.11 l f ?,G$:1bH,o. 16.399 44.877 Td (1) Tj jd;36ssNJZJEHC)WW*Q-e6,'lml\*P 11.96 0 0 11.96 0 0 Tm The definition of addition and multiplication modulo follows the same properties of ordinary addition and multiplication of algebra. Q q 50.740 40.269 Td (6) Tj 1.070 0 Td (Z) Tj -26.345 -1.166 Td (elemen) Tj Tj /F4 1 Tf KD:6"! 0.462 0 Td (w) Tj 1.713 0 Td (d) Tj 2.297 0 Td (to) Tj "@jMuM`@Rr]qC9,Lc*rKI5*8bqXMf$.jkB>JB0jubNK*V? /F3 1 Tf 1.322 0 Td (4) Tj /F4 1 Tf U42@fFl!aLg*Kr8RGi"tgNpO"\?h>W]QXbXZTKJbV+"djaap,K]F:(/KR^Y).u4V1DHUq492BBN:ADrj^X'&:ikY=WgVLjI.,,oT^,&/l /F6 1 Tf /F4 1 Tf 11.96 0 0 11.96 0 0 Tm /F5 1 Tf Q q BT /F11 1 Tf /Length 90 0 R 1.812 0 Td (=) Tj /F8 1 Tf /F3 1 Tf /Length 92 0 R Q q ]!u(]B)qr^=d@1 2.339 0 Td (function) Tj Tj Suppose we want to find the equivalence classes of mod 3. 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