1、1 M mu v ip 4/ s1“1.l aab(b 6= 0),i q, a = bqSa ?$b“,:Tbja. aSb , bSa (y ).b 6= 1,5bSa .a ?$b“,5:Tb-a.Tatjb, at+1 -b, t 2 N,:Tatkb.2.1“Bte(1)bj0, 1ja, aja(a 6= 0).(2) bja, a 6= 0,51 6jbj6jaj.(3) cjb, bja,5cja.(4) bja, c 6= 0,5bcjac.(5) cja, cjb,5cj(ma+nb)(man 2 Z).(6) kPi=1ai = 0, b ?“a1, a2, , akk1
2、,5b ?“ 6B.21.l!m , ab$m“ M,5ab m,:Ta b(modm).2.(1)a b(modm) , mj(ba).(2)a b(modm) , b = km+a(k 2 Z).(3)a a(modm).(4) a b(modm),5b a(modm).2(5) a b(modm), b c(modm),5a c(modm).(6) a b(modm), c d(modm),5ac bd(modm), ac bd(modm), an bn(modm).(7) ac bc(modm), (c; m) = d,5a b(modmd ). |(c; m)V UcmKv .+Y,
3、(c; m) = 1 H, ac bc(modm),5a b(modm).3. 1 m F“, B“S1 m (S1 m : ). $m“ ? 0, 1, , m1m f, , “ V m1“sm0“: A0, A1, , Am1.Ai = fqm+ijm , q 2 Zg, i = 0; 1; ; m1.Ai(i = 0; 1; ; m1) m1Si=0Ai = Z,m1Ti=0Ai =?.4. :“Vmm A0, A1, , Am1, B Ai |B ai,5a0, a1, , am1S mB :“(e m“).Ke m :“ 0, 1, , m1,9S mKld“.A mM? mB“.3
4、 1.Bv1 , T1 T“1 9 .yN, “Z+ = f1gSf gSf g.2.v1 ,Kl B .3. aKl vpa.4. k.5.i“ Tf(n) =mPi=0aini, P i1 n, f(n) .6.:(Wilson) p sA1Hq (p1)! 1(modp).4y s1.y s ( Bs )3Bv1 ?sy T, O Tty vlv (My T),sZE B.2. n(n 1)Ss Tn =mQi=1pfiii . pi , fii , i = 1; 2; ; m.3. !d(n) = Pdjn1V Uv1 n , nSs Tn =mQi=1pfiii ,5d(n) =mY
5、i=1(1+fii).4. ! (n) = PdjndV Uv1 n , nSs Tn =mQi=1pfiii ,5(n) =mYi=1pfii+1i 1pi 1 .5.n!Ss T,y pZ 1Pr=1 npr. :|xV UVxKv .5 1. Kv (1) cja1, cja2, , cjan,5ca1, a2, , an .a1, a2, , an KvBa1, a2, , anKv .:T(a1; a2; ; an).(2) a1, a2, , anSs Ta1 =mQi=1pfiii , a2 =mQi=1pflii , , an =mQi=1pii , pi , fii, fli
6、, id , i = 1; 2; ; m,5(a1; a2; ; an) =mQi=1ptii , ti = minffii; fli; ; ig.(3) Ta b , * ab “b “M.(4) Ta b ,5(a; b) = b.(5) !ab H1 , Od = ax0 +by0 ax+by(xay ) Kl,5d = (a; b).(6) ab “ Kv “M.(7) !m i ,5(am; bm) = (a; b)m.(8) !n abB ,5an;bn= (a; b)n .(9) ! ab(a b) Ta = bq +r, 0 6 r r1 r2 r3 #ri(i = 1; 2;
7、 ) d ,5B BQH, n+1Qrn+1 = 0.rn 6= 0,5(a; b) = (b; r1) = (r1; r2) = = (rn1; rn) = rn.NE V p(5) ax+byKl d = ax0 +by0.2. Kl (1) a1jb, a2jb, , anjb,5ba1, a2, , an . a1, a2, , an KlBa1, a2, , anKl .:Ta1; a2; ; an.(2) a1, a2, , anSs Ta1 =mQi=1pfiii , a2 =mQi=1pflii , , an =mQi=1pii , pi , fii, fli, id , i
8、= 1; 2; ; m,5a1; a2; ; an =mQi=1prii , ri = maxffii; fli; ; ig.(3)a1, a2, , anKl B .(4)a; b = ab(a; b).6 an l 0 1. (1) (a1; a2; ; an) = 1,Sa1, a2, , an(9S ).n ( ).+Y, 1 ;M # ;M # ; p, p ?“a,5pa.(2) (a; b) = 1,5(ab; a) = 1, (ab; ab) = 1.(3) (a; b) = 1, ajbc,5ajc.(4) ajc, bjc, (a; b) = 1,5abjc.(5) (a;
9、 b) = 1,5(b; ac) = (b; c).(6) (a; b) = 1, cja,5(c; b) = 1.(7) (a; b) = 1,5(a; bk) = 1.(8) a1, a2, , am Bb1, b2, , bn B,5(a1a2am; b1b2bn) = 1.52. x f l:lm Om S x (Euler)f ,:T(m).m =nQi=1pfiii ,5(m) = mnQi=11 1pi.pi , fii (i = 1; 2; ; n).m H, (m) = m1.:(1)(m) f ,(a; b) = 1,5(a)(b) = (ab).(2) p ,5(p) =
10、 p1, (pk) = pk pk1.(3) !m = pfi11 pfi22 pfikk ,5(m) = m1 1p1 1 1p21 1pk.(4) !d1; d2; ; dT(m) m ,5T(m)Pi=1(di) = m.3. x n l (1) x !m 2, O(a; m) = 1, (m) x f ,5a(m) 1(modm).(2)n (Fermat)l !p , O(a; p) = 1,5ap1 1(modp).:n l x m H+ .4. 0 !m1, m2, , mk k .5 TFx b1(modm1),x b2(modm2),x bk(modmk)Bx M01M1b1
11、 +M02M2b2 +M0kMkbk(modM).M = m1m2mk, Mi = Mmi, i = 1; 2; ; k, M0iMi 1(modmi), i = 1; 2; ; k.: 0 S : .7 1. B ?$2“,5 ; B $2“1,5 .6 “ “ 2 .2. () , () .i () .B B () . M .3. i . i B ,5 .8 Z 1. a ,5a2Sa Z .2. Z ? 0, 1, 4, 5, 6, 9.3. Z E .4. 5 Z , E 2, .5. TB Z 6, * E .6. Z ?$4“; Z$4“1.7. Z$804; Z$8“1.8. B
12、 ?$3“,5 Z ?$3“; B ?$3“,5 Z$3“1.9. B ?$5“,5 Z ?$5“; B ?$5“,5 Z$5“+11.10. Z MF, T B , MF, B ,B ? 0, 1, 4, 7, 9.11. M # Z W V ? Z .12. Z ,i OQV 9 .13. T p B Z , * p29 Z .9 V“+1.B ?$2“ sA1Hq .2.B ?$4“ sA1Hq ?$4“.3.B ?$5“ sA1Hq 05.74.B ?$3“ sA1Hq 3 ?$3“.5.B ?$9“ sA1Hq 3 ?$9“.6.B ?$11“ sA1Hq 3 3 ?$11“.7.B
13、 ?$10n 1(n )“ sA1Hq ,F n, ?$10n1“,AA = 10x + y, y 2 f0; 1; ; 9g,5(10n1)jA ,(10n1)j(x+ny).N V A ?$9, 19, 29, 39, “.8.B ?$10n + 1(n )“ sA1Hq ,h n, ?$10n + 1“.AA = 10x + y, y 2 f0; 1; ; 9g,5(10n + 1)jA ,(10n+1)j(xny).N V A ?$11, 21, 31, 41, “.10 E: E1. A EV UA =nPi=1ai10i, ai 2f0; 1; ; 9g, i = 0; 1; ;
14、n1, an 2f1; 2; ; 9g.2. AnQ A nQ ,An an0(mod10).3. An 4 C.4. A 3S(A) =nPi=0ai1 9,A nPi=0ai(mod9).5. A 3S(A) =nPi=0ai S(A+B) 6 S(A)+S(B), S(AB) 6 S(A)S(B).6. ab id ,512a 5bl Z 7 T K.7. 1n Kl Z 7 T,5n = 2a 5b, aabd .8.1n El Z 7 T,vn1.9. (n; 10) = 1,51nr, r 10r 1(modn)Kl .11 k: E1. !k 2 B (),5 B E A VBk
15、V U, V / T: A =d0+d1k+d2k2+dnkn =nPi=0diki. di 2f0; 1; ; k1g,i = 0; 1; ; n1,dn 2f1; 2; ; k1g.2. AkV U V:A = (dndn1d1d0)k.3. !BBl ,5B VBkV U, V / T: B = d1k1+d2k2+dnkn + di 2f0; 1; ; k1g, i = 1; 2; ; n; 8: BKl ,5 TK; BKl ,5 TK.12Z1.=BQZax+by = c(1)Zax+by = c(aabac ) sA1Hq (a; b)jc.(2) (a; b) = 1, O(x
16、0; y0) Zax + by = cBF ,5x = x0 + bt, y = y0 at(t ) Z .2.Zx2 +y2 = z2 (1) x = a, y = b, z = c(aabac ) Zx2 + y2 = z2BF, O(a; b) = 1,FZBF.(2) x = a, y = b, z = cZx2 +y2 = z2BF,5ab B , c .(3) !x = a, y = b, z = cZx2 + y2 = z2BF, OLa ,5i mn,m n, (m; n) = 1, Om 6 n(mod2), Pa = 2mn, b = m2 n2, c = m2 +n2.(
17、4) a = 2mn, b = m2 n2, c = m2 + n2,5aabac x2 + y2 = z2BF; Tm n 0,(m; n) = 1m 6 n(mod2),5aabac ZBF.3. :(Pell)Z(1)Zx2 dy2 = 1(d ),S :Z.(2) d | I , x = 1, y = 0 :Z,F :Z O.(3) !d 0 Bd Z ,5 :Zx2 dy2 = 1 k .(4) !n 0, (x1; y1) :Zx2 dy2 = 1B, !xnyn/ Tl(x1 pdy1)n = xn +pdyn,5(xn; yn) :Zx2 dy2 = 1B.13 US“,a:U
18、S ( S,9. , Vl bWUS“.1.H T eH(1H), S,H = N,HH L,5S = N + L2 1.2.Z =(1)H ( USZ, T =c, Kv 1.(2) =cZ ,Kv 2.9(3) =cBKvZ 4.3. =5!A(r)V U ux2 +y2 6 r2 , r L ,5A(r) = 1+4r+4X16s6rpr2 s2A(r) = 1+4r+8 P16s6 rp2pr2 s24 rp22., xV UVxKv .N,r sv H, ux2 +y2 6 r2 A(r)r.4.i .5.n 5 H,inH.14f x1.l!x 2 R,5xV UVxKv .2.f
19、 x(1)y = xl L “R, “Z.(2)x = x+r, 0 6 r nPi=1xi.(7) L x1; x2; ; xn nQi=1xinQi=1xi.+Y, x# nxn xn, x npxn.(8) L xayhyxi6 yx.(9) !n ,5hxni=xn.(10) x,x = x;d x,x = x1.(11) mn,vmn hmni.10(12)f fxgl L xBl s,fxg = xx.y = fxg /Bt:(i)fxg2 0; 1).(ii)fxg 1Kl f .(iii)fn+xg = fxg(n ).(13) !p 2 N, 2j(2p)!KvM = 2p
20、1.(11)M =2p2+2p22+2p23+ = 2p1 +2p2 +2+1 = 2p 1.15 1. l(a; m) = 1,Kl , Pa 1(modm), Oak 6 1(modm), 0 wL“, Kv.1. 2. 6 BnH“,nHKl.1. 2. 7 Be wL“,Kl.1. 31 (1) (Menelaus) BL4ABC HBCaCAaABLsYXaYaZ.5AZZB BXXC CYYA = 1.(2) m I !XaYaZsY 4ABC HBCaCAaABL . AZZB BXXC CYYA = 1,5XaYaZ L( L).(3) (Ceva) !P4ABC =B,LAP
21、aBPaCPsYHBCaCAaABDaEaF.5AFFBBDDCCEEA =1.(4) I !DaEaFsY 4ABC HBCaCAaAB . AFFB BDDC CEEA = 1,5ADaBEaCF LB( ).(5) (Ptolemy) HABCD = H,5AB CD +BC DA = AC BD.(6) I HABCD AB CD +BC DA = AC BD,5 HABCD = H.(7) AC BD,| Hq O HABCD= H.(8) (Simson) !4ABC iBP HBCaCAaAB gDaEaF.5DaEaFBHL( L) .(9) I !4ABC BP HBCaCA
22、aAB gDaEaF. DaEaF L,5P4ABC .(10)n (Fermat)54ABC Kl|n F.4ABCKvl120 H,F1 HBCaCAaABf (120;4ABCKvv120 H,FKv.(11)4ABC ZKl|G.5 v(Carnot) G4ABC, P4ABC iB,5PA2 + PB2 + PC2 =GA2 +GB2 +GC2 +3PG2 GA2 +GB2 +GC2; D%(Leibnitz) T G4ABC, P4ABC iB,5PA2 + PB2 +PC2 = 3PG2 + 13(a2 +b2 +c2), aabacsY4ABC HH.(12)4ABC = H
23、Kv|G.1. 4+ M1. 4. 1M 1fi).fi = H,Q M,.(3)M(Q M): mFM1LlmF0,“MS1Ll(Q )M,:U(l).1. 4. 2M M15 1k).(2) M: !OB, k(k 0) , _,mF iBP, LOP O , L iBP0,OP0 = kOP,PP0MSO a a 1k M,:S(O; ; k).1. 4. 3(1) TmFmF0 B MM,5iM,V QM, V PFMF0. l1l2 MZ_ u, i W Kv.1. 5. 5: TmF B nf G1, G2, , GnB,5mF$G1, G2, ,Gn; T 8“bG1, G2,
24、, Gn FB ?$n B c,5G1, G2, , GnF.1. 5. 6(1)F F.(2) G1 G, G2 G1,5G2 G.(3) G1 F, G2 F,5G1 G2 F.(4) T G ? uF,5S(G) S(F), S(X)V uX ,/.1. 5. 7 (1) jnHM i OG HQM,5 H l *jnH B ?$H v *;(2) T ? sBO, P“F BO V r,5FA V$Br .(3)AaB, fi, “F BPAB, OAaB jAPB fi,5“F ?$AB?acfiB G .(4) TGF u, OS(F) S(G),5GA ?F.(5)Bd“F ?$
25、ld“G .(6)1Lb tL ,tL v1,5tL H .(7) T1 b t,tv2,5t .(8)Lnf , sY S1, S2, , Sn, 3 B S u, TS1 +S2 +Sn S,5 f ? 3.(9) ! SmGcmG1, G2, , Gn, sYS1, S2, , Sn. TmGB$Gik ,5S1 + S2 + Sn 6 kS; S1 + S2 + Sn kS,5G iB$fGigk +1 .2 8+ 2. 1 : i O i LB =nH L,#M # H LW s Fm,S .F LS ,t L SS 17,M # W sS ,M # FS ,M # F= S = . v , l360.2. 2 x (Euler) : e 8 V, F, E,5V +F E = 2.3+ 3. 1LZ(1)EL T: xcosfi + ysinfi = p, p_L 1) ,Vn Bk- V . “ nk.(3) Ms !n VskF,BF 4NM,FW M. !kF GQn1, n2, , nk(n1 +n2 +nk = n),5n Ms , n!n1!n2!nk!.19(4)
