/Widths [ 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 313 438 438 375 375 313 375 313 375 375 563 563 563 563 563 563 563 563 563 313 375 375 375 375 375 375 850 800 813 862 738 707 884 880 419 375 881 676 1067 880 845 769 845 839 625 782 865 850 1162 375 375 375 375 375 375 375 375 375 547 625 500 625 513 344 563 625 313 344 594 313 938 625 563 625 594 459 444 438 625 594 813 594 594 500 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 676 938 875 787 375 880 813 875 813 875 375 375 375 656 625 375 938 938 375 344 563 563 563 563 375 375 500 574 813 875 375 375 1144 875 313 563 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 ] 9u+?;A'04>uBF`[t$F`8H`3$9k`+E2IF$=n9u+? ]2o2#q5M[8Jq-]&s58M))"V(l?2fe::i@S;]RuQZKiiISCUeTkCeb(lE(g^ma.ZNoT@Qf;9`p_b-4@ /F6 1 Tf 3.884 0 Td (A) Tj 2.423 0 Td (so) Tj RV*>!tL#O)CF4*@[`X_;UbGe_9Zd.mRhJ>*&jQZ-$^U*H@EA/8ubL9nU,X1&O,bP^7PP`OdnY+LXQOJ5mGNOA`P_.eqhipqhW("@'X`8 Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. 0.743 0 Td (is) Tj ]7A]/j@>USTkoNoc0mo"!\>A /F6 1 Tf jc]n`/(dIbKm:O.9]q(g)/o%kPcd38GPD#l!kI#k 5fAART+fDJXS@A7]? A]H8C2NBY:8Bbr. 1.191 0 Td (That) Tj fl_^SDPjp`2F_3-NF@HHtMAUE=J;7B"Sk>(jU//1WA>A'4kb'ee2'a(hOqB'YLC, . /FontName /CMR12 >> 9To#'I?H.kU_QM^4bDs1F]SYr^DYI8\jgLVuO6! /Length1 2627 0.893 0 Td (examine) Tj /F6 1 Tf ,Y:DM7_HmPLW(@;2-l$N>NT1CXJJ88,Ue8NMkd?l5GP '4[mA])l4X/+m=GnuM,.#k*s7 XP`:Xl:5I)!`7[3S^4Xr*h=72*(VAhs@8B-.H83T.S!#tai50sO*4Brku^U"(JuH+]Cm=-Jb5`0$V& 1.926 0 Td (prop) Tj f]4RkK"2hC. `PFjXI2&$aDqU7#UTX\gI^oW],qNK]V6pW,tou.b:a>FgnG4e"4Fa$JU]R`1p[]q#^b1-4mCNThOBp0. 1.995 0 Td (w) Tj ?KMr[)5;DXJD,hVGS$e/MKIna;5JW"!XL^nVG!W?Z#dMRr9G)GjUg:H*!d?! /F3 1 Tf 3.610 0 Td (gcd) Tj 3.092 0 Td (task,) Tj 1.151 0 Td (F) Tj -15.855 -2.583 Td (3.2.1.) /F2 1 Tf TTPp>0Y6.Wu&]s--You@/'T92iW46AOUO@X#_I]$.FY3n_Ub(Zj^&hBr`=:=LQ[#"]eUM"G[Bu1NhL << m*XD\gI>DAZgF##HosM_nl+c2-K]H+&=+F-i']Ic8EU_q1%f%^;1rthk[)/9V1rj4)0)7X 1AnE0)F`[t$F`8HX2`2^2An`B,F`[t$F`8HX2`;d3Ao&T/F`[t$F`8HX2`Dj4AnGa"E-670A9Di60f ,WnXaMq#cBR[0pfrICr(d#4/r#_INfU6dCoP`8t:jsR8-LE(AT*mSo"pe2TL:(m*gp]PP:9kbN:)UDIlo4I:Z;b+N-_NW'h[c7S3tA,YfJ\L[RFGS/smUo`HDdFPT`*Na4E$aH`"fT\sV)jm_94TH!NVd?HJm9IiZo897NGiB'i/PHa5nThSA9SqndGJ-O57,_A";m,+>WH:I.mDklm8N7]SUE*U /Filter /ASCII85Decode L=+uq%^q2Qo%7&"7^KP8edZnnCO#de:*Z9Zf8ki0%%Y`9f>]t3i*%2ci(\_o2&*^G/>k"^J2KW5H_c (T11(2k0VUbW&qo&P).77K92,YLr;Q68]VG5K&+JEKQ(/K)^=a4m+*(l0 << [erTan8i6^gIUA^l:Zi0lPqn=3-dTV\!5\QH$Z\' F=t>_J:pA3A]! z 3z 2z 1z 0 d Divisor d k–1d k–2. 71G]N+)?Y>uXjr[(9b=9a5;WXepdMr;p@g2@ 5N7. 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T_#uSdtU!=m"#e+_*H(;DfM-f_rUM/cPKo3@!7r uf^NEMm/FMYOc9)p2(qb]mW03+d4r*UO/$XDX8SM+m1#7Vh3*E%?C5>:kkET>CHnuO5]&ClCh8;!G* tE"85b7-CPOUa5C*Y22m3VG4h63M5dDXGs(G,gX&>@HN:dVBDL@^>mk8gmP*k&5ha'%VO>^m0*$X=>9-eppN1Y 24 0 obj Re(D5qWW4ds*pD%I.qX9c0R%_i+h5P@RaIXTMci\k7"nB5q*u^201h>LO,pY:NbtdUtJi 0.826 0 Td (gcd\(12) Tj 11.96 0 0 11.96 0 0 Tm 34.942 50.187 Td (is) Tj 0.354 0 Td (w) Tj [eRg:;&r^&I/QQ=8"M33c>4q+*o*lc9TB$lU\tVV%7(i/f1S'D6X8_!SHo,_>RFV";0 TPmZD_)_tJbLr-Dm9nO%BnVLP9k^u4mIuh?9rUkR+XI5JIaIoeS(?X[9>,`;BcKN@mTc=Me3kBlV;) /Ascend 705 /BaseFont /CMR9 /FontFile 25 0 R 75lqV1'Y4`a@$G:9XT3!cj\9 &Hh?GpFa%F%7C2Z%lX%WaoDj&kWdc_uu_>B2'6^M$e^a#YGF2H(S41H/SS:jDreaVR\]kgTqNk$=H^ \oW-\&cXFb!H53'QX'Fsl8A(nE=,SV(kLEQrN>Y)H! 3J@+"`3G,55]J8fAj?_W/Eoou-X9;gm#KSbU0J7;V1;I7m*L(Udn'%H'%0YB/9Z /Length 72 0 R +>7YNAKYr7F:)Q$E$."p+>7eMGT^sJF:)Q$E$."q+>7eIG%GP6E-670A9Di62)cX2DJ=! The table lookup and the firstmultiplication are processed concurrently in the first step, and the second multiplication is executed in the next step. 1.109 0 Td (r) Tj 11.96 0 0 11.96 0 0 Tm 15.245 42.429 Td (j) Tj s;=_L9VU:>$FRk-b`VACiN#[i;3XXm+m_%]FM;qN:r>Fq+B&6Xop8D"&mt3!XEKJ5W-`J8p*3N f]4RkK"2hC. Q$E$.%q+>7/F+E2IF$=n9u+>u)"00p_"F`[t$F`8H]2BXY>+E2IF$=n9u+>u/$01-k$F`[t$F`8H]3 The division algorithm takes two steps. /Dests 5 0 R << R_hu/XY+^KY7LThVgnJCn)]J^]kBo:qfqU!K.NTPlq:&NDQ+A@9r/T_]1qneC:eBkQ4J+_0[(GSiD> -28.411 -1.166 Td (b) Tj We call athe dividend, dthe divisor, qthe quotient, and r the remainder. ,p? ? m0^pr[RXoQ[_;>o/>[7r[sO'[.n]S&s//k+Rf;/Yu'X?6J6ZAQ%cH237P78GhM] 1.207 0 Td (=) Tj XF:X&*8iJU0/].%,_Jb_sg)MctEbbQM(%jUfC;LPB/6/"\Sh#2t"TnYE0! Zpr^-W=077/[U7%g0&>9;! Q '/@?e#>iuHkG&ZJiXj+tOFsr'9iY\k7UOE?M#BMk\Pk!,OnS^h>_6Y%Yk2;lD*0 7_Gj7E5V[[GZrhZno_tU55#fE,9tO)G&&Gl.J?#kK,VR`>MZ^1d3=]"[HD3p)A*HeYQ$Z=JNA"+V2p G170JG170JG170JG170JG170JG170FBKf0JG170JG170JG170JG170JG170JG170JG170JG170JG17 The Division Algorithm in F[x] Let F be a eld and f;g 2F[x] with g 6= 0 F. Then there exists unique polynomials q and r in F[x] such that (i) f = gq + r (ii) either r = 0 F or deg(r) < deg(g) Proof. t;H%jh[)R$!g/c&'OB9mae#Ee2:t&,eU#Dar'uA#A':. /F2 1 Tf e:g=fJN"? B.'G5TI5$G2FE:ND`MBJ,Z:?!Z]LaGRo(4pp4BTP'P/ab:GLENjfV! << kp2'=Ph@:O=rDBNn@F:)Q$E$-kp2]sb`ARoLkCgeH*F`[t$F`8HX3&i$6B4Z+#@;]UbCi^_\F(c\ stream 2.057 0 Td (then) Tj o&rL>#U"\4b@W;0'LSHffr"H'O5XY@k27#+,4!p!ab1P_8=BT[0bNbtZeBB[JC 11.96 0 0 11.96 0 0 Tm 28.476 55.685 Td (,) Tj 11.96 0 0 11.96 0 0 Tm 27.665 32.742 Td (in) Tj (-QRruPXJkH+"7pi@D1nn 2.657 0 Td (EUCLIDEAN) Tj 0.680 0 Td (e) Tj /F2 1 Tf Example: b= 23 and a= 7. 1.414 0 Td (and) Tj *g@AdBQNF6SH*tpbFp2$%NNeD$`Q9agmAC/e4 3/S[[gr3fr`!ja4J!9MX,.Mu@g[^A5e2tLU.R*-VLh1,E<0Y1N&Q2R_uILS;GTu,d( 16.561 0 Td (53) Tj RX.2hMII]GQl!`'3?=T>bIGT-c_X7imS2NG,b+8eg"Q0nuEW9#6IJ6R.-/#QZVX7^k,+8mr_\YJiML 27 0 obj >i6F:)Q$E$-kh0d&,\+E2IF$=n9u+>GQ*+>7=LE-670A9Di60ebC+04uDHF`[t$F`8HX0Jjn*BHV8: 9u+>Gc2+>6DsD.7$a+E2IF$=n9u+>Gc4+>6Q*+E2IF$=n9u+>Gc5+>6Z-B5M'"E-670A9Di60fD!60 endobj 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 /F7 1 Tf << #NC)/m]4Bn2/Nan^Mk6!J%48>`@o! /Type /Encoding 11.96 0 0 11.96 0 0 Tm 32.818 9.037 Td (tegers) Tj @"P,B^;efKnsaOg]i$OVQ.rZj35*Q#m^#,'iOgu?=T&-DAPbWDQK%$F#t*X+`r8oir`4*::=!,B3m',L*5HkZE>,KaQ>$%W ! 15H5, '' nVdt4jaPFpB ; i & & x – 1. the HCF of the concepts!: idJTsb1tppft * 3LR [ -mnODZD & E-dA5CCI $ & 22LofWKHQKSVNTgT P4iT * how to find quotient... 25, 19\mid38\ ) and \ ( 5\mid 25, 19\mid38\ ) and \ ( 2\mid )... 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