§1.3CPCBBBATBZ7
CNCMD9A7
(a(a,b),b(a,b))=1.
AIAPAGAQAGBFAHCLCLCQA7A6DJA3B1D3AGBFAHCLCFC8ANARAHA9
D9BU1.2.12.DFa1,a2,...,ak∈Z(k≥2),d∈Z.BJdA8a1,a2,...,akCLA8A6CF,D6
d|ai,i=1,2,...,k.
D0B6CUBLATCLBFAHa1,a2,...,ak,BJAYBGCLANARAHBSBJCFC8BCBLa1,a2,...,akCLCFC8ANARAHA7C6
CI
(a1,a2,...,ak).
D0AGa1,a2,...,akCUBL0CLCLAAA7AYCTCCCFC8ANARAHBL0.
D6(a1,a2,...,ak)=1,B2BJa1,a2,...,akBKAQA9D6a1,a2,...,akBSCZA1AQAGAHA0BKAQA7B2
BJa1,a2,...,akAQAQBKAQA9C9CXA7AQAQBKAQATBKAQA7CEDDBJBOATA9
C6AGBNB5CMCQD3AGBFAHCLCFC8ANARAHCLBUB5AZCXBLCQAQAGBFAHCLCFC8ANARAHCLBUB5A9
ASB81.2.13.C3ai∈Z,i=1,2,...,n.AT
d1=a1,(dj?1,aj)=dj,j=2,3,...,n.
AA(a1,a2,...,an)=dnBRD0A8s1,s2,...,sn∈ZC7D8
a1s1+a2s2+···+ansn=dn.
BJB81.2.BHBKC6ASAHB5A9
1.CQ(51425,13310)C1BFAHs,tA1CK(51425,13310)=51425s+13310t.
2.BHBKBNB51.2.13.(B4A4ABD0nABAHD1AZBTDBB4ALABBFB41.2.7)
3.CQ(353430,530135,165186).
4.DFanxn+an?1xn?1+···+a1x+a0A8BFC5AHD3CGA3A9AFDFu,v∈Z,(u,v)=1CK
v
uA8A9
D3CGA3CLAJA9BHBKv|a0,u|an.
5.DFa,bA8BFAHA7mA8BGBFAHCKm|a,m|b.AFDFC5AZBFAHs,tA1CKm=as+bt.BHBK(a,b)=m.
6.DFa,b,nA0BLBGBFAHA9BHBK(an,bn)=(a,b)n.
§1.3CEBPA3COB2
ASCWB2B4BFAHCLCFCIANAPAHA9AICFC8ANARAHCLCQDED9A7C7A3B1AQAGBFAHCLCLAAA9
D9BU1.3.1.DFa,bA0BLDJATBFAHA9BJd∈ZBLa,bCLA8CGCF,D6a|d,b|d.BJa,bCLBGANAPAHBJ
CFCIBCBLa,bCLCFCIANAPAHA7C6CI[a,b].
C6BJCLBNB5A8C9CXCLA9
8B2CDCKCLAXBNBQ
ASB81.3.2.C3a,b∈ZB2CNCVBBA9AA
(1)[a,1]=|a|,[a,a]=|a|;
(2)[a,b]=[b,a];
(3)a|b=?[a,b]=|b|;
(4)[a,b]=[|a|,|b|].
DAAGBNB51.3.2BMBTA7CQAQDJATBFAHCLCFCIANAPAHCLBUB5A6AZCXBLCQAQBGBFAHCLCFCIANAP
AHCLBUB5A9C6BJCLCTAJCSA4ARCFC8ANARAHBCCFCIANAPAHCLAUC5A9
D9AJ1.3.3.C3a,bCAAF?CFA9AAa,bD9CLDHAFA8CGCFCA
M=
braceleftbigg
k·ab(a,b)|kCAAF?CF
bracerightbigg
.
C9ARC9CXA7BZBEMBSCLAHCVA8a,bCLBGANAPAHA9DDBJA7DFBGBFAHcBACDa|c,b|c.B2C5AZBGBF
AHx,yA1CKc=ax=by.AGA8
a
(a,b)·x=
b
(a,b)·y.
ACBFB41.2.11BI(a(a,b),b(a,b))=1.DEBNB51.2.9(4),AD
a
(a,b)|y.C6y=k·
a
(a,b).B2c=k·
ab
(a,b).AT
c∈M.square
BDAP1.3.4.C3a,bCAA1BB?CFA9AA
[a,b]=|a||b|(a,b)
BRa,bD9BXDEA8CGCFB2CA[a,b]D9CGCFA9CNCMD9A7BZ(a,b)=1,AA[a,b]=|a||b|.
CXBEBNB51.2.10,ADDJC6CXB1A9
BDAP1.3.5.C3a,bCAA1BB?CFDFmCAAF?CFA9AA[ma,mb]=m[a,b].
AM1.3.6.DFADAQBGBFAHBJBCBL667,D5AYBGCLCFCIANAPAHBWDJCFC8ANARAHCMAG120.CQBDAQAG
AHA9
C9ARDFBDAQBGBFAHBLa,b,a≥bCKa=(a,b)a1,b=(a,b)b1.B2DEBFB41.3.4BI[a,b]=(a,b)a1b1.
ACDHBIBACJBI
(a,b)(a1+b1)=667,a1b1=120.
C4A1CH
667=23×29,120=23×3×5,
AD(a,b)=23,a1=5,b1=24BU(a,b)=29,a1=8,b1=15.AGA8BDAQAGAHBL115,552BU232,435.
square
AICFC8ANARAHDED9A7CFCIANAPAHDBA6BFAXCHD3AGDJATBFAHCLCLAAA9
D9BU1.3.7.DFa1,a2,...,ak∈Z(k≥2)CKA0B6BLATA9BJd∈ZA8a1,a2,...,akCLA8CGCF,D6
ai|d,i=1,2,...,k,
D5a1,a2,...,kCLBGANAPAHBSBJCFCIBCBJBLa1,a2,...,akCLCFCIANAPAHA7C6CI[a1,a2,...,ak].
§1.4C2BYBIAUB3BN9
ASB81.3.8.C3a1,a2,...,akCAA1BB?CFA9AT
m1=a1,[mj?1,aj]=mj,j=2,3,...,k.
AA[a1,a2,...,ak]=mk.
BJB81.3.BHBKC6ASAHB5A9
1.C5AT[391,493].
2.CQ[209,665,4025]B4ALBK
[209,665,4025]negationslash=209×665×4025(209,665,4025).
3.D2ADCCDIAQAGBOB3BKCEC8CTA7CCB3AD437BOA7DIB3AD323BOA9CCCLBQDEBOAIDICLBQDEBOCE
CTBZBICHAYBKCECTBZA7BPDBAHDAC5C3BX?
4.DFa,b,cA8AQAQB6BCCLBGBFAHCKAQAQBKAQA9AFDFD7AGAHBSCZBDAQAGBJBCA0A6ARCOD7AGBFBWA9
CQa,b,c.
5.DFBQD7BQAHA8CCAHBQAHB7BJBCCL12APA9CQBDAGD7BQAHA9
6.DFa,bA8BGBFAHCK[a,b]=(a,b).BHBKa=b.
7.DFa,bBLBGBFAHA9BHBK(a+b)[a,b]=a[b,a+b],(B4A4ABABBFB41.3.4BCBNB51.2.5.)
8.BHBKBNB51.3.8.
§1.4B6B1AACPDBAL
ASCWD1DCAQAHA7BEAHCMAABZA7AZC2BVBXDACKANATAFBVASCTAJB4A3B1BDDECTAJCLDECKA9ABA9
ABC7AIBTAQAHCLAABZA9
D9BU1.4.1.DFa∈Z,a>1.BJaA8CICF,D6CCBGARAHBOAD1C1CCASDGA9A2B2A7BJCCBLAICF.
CBAB1.4.2.AQAHDBBJBLBRAHBUB6A6ARAHADAQAHA0BLBGBFAHADCUB6BGBFAHA1BL1,AQAHBCBEAHD7
AIA9
AM1.4.3.DFnA8BGBFAHA9B2
(n+1)!+2,(n+1)!+3,···,(n+1)!+n+1
A8APCZCLnAGBEAHA9
D9AJ1.4.4.AICFaD9CW1DDCRD9ASD4AFA6CFCACICFBRCNCPAD
√a.
C9ARDFaCLBW1DJBGCLCFCIBGARAHA8pCKa=pq.D6pBLBEAHA7B2C5AZp1,p2∈ZA1CKp=p1p2
CK1
A8BEAHA7ADqnegationslash=1.ACpCLCFCICSBIp≤q.ATa=pq≥p2.C2p≤√a.square
BDAP1.4.5.C3aCAD2DJ1D9?CFA9BZCNCPAD
√a
D9CICFB2CNBJ?CWa,AAaCACICFA9
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