Giáo trình Đại số sơ cấp

TRUCiNG DAI HOC HUNG V U d N G

NGUYEN TIEN MANH (CHU BIEN)

HOANG CONG KIEN - TRAN ANH TUAN

NGUYEN THI THANH TAM - NGUYEN V AN NGHIA

NHA XUAT BAN DAI HQC THAI NGUYEN

iUYEN

LIEU

T R U O N G DAI H O C H U N G VU'ONG

NGUYEN TIEN MANH (Chii h ic n )- HOANG CONG KIEN,

TRAN ANII TUAN, NGUYEN TIH THANH TAM , NGUYEN VAN NG H IA

g iA o trin h

DAI SO SO CAP

NIIA X U A T BAN DAI H Q C THAI NGUYEN

NAM 2017

D H T N -2 0 1 7

M LC LUC

Loi noi dau 6

C huong I. ( AC TAP HOP SO 8

1.1 So t\i nil en............................................................................................................. 8

1 .2. So nguyen.............................................................................................................. 11

1.3. So huu t ................................................................................................................. 19

1.4. S oth u c................................................................................................................... 21

15. So phuc................................................................................................................. 27

Bai tap............................................................................................................................ 30

Chumng 2. fcAI SO K [ X ] VA SO HOC TRO NG K [X] 35

2.1. Dai so / [.V ] ........................................................................................................ 35

2 2. So hpc t ong AT [ A'] ............................................................................................ 50

Bai tap............................................................................................................................ 63

C huong 3. (.HONG DIEM C lIA DA T IIU C 67

3 1 Khong dem cua da time.................................................................................... 67

3.2. Cong thtc npi suy................................................................................................ 80

3.3. Mpt vai oai da thirc dac bict........................................................................... 84

3.4. Da thirc loi xirng................................................................................................. 91

3.5. Bai toanxac djnh da thirc................................................................................ 104

Bai tap............................................................................................................................ 116

Chirong 4. IHAN I lllTC IlC'lJ I I 125

4 1. Truang ]han thirc huu ti K [,Y J ................................................................... 125

4.2. Mot so bai toan ve phan thirc............................................................................ 131

Bai tap............................................................................................................................. 152

C h u on g 5. HAM SO VA DO THI 154

5.1. Khai ni^m ve ham so va do thj........................................................................ 154

5.2. Khao sat mpt ham so bang phuong phap so cap......................................... 156

5.3. Cac phep bien doi do thj.................................................................................... 161

5.4. Khao sat so cap ham so bac nhat va bac hai................................................ 169

5.5. Khao sat so cap ham phan thuc..................................................................... 170

5.6. Khao sat so cap ham so mu va logarit......................................................... 171

5.7. Khao sat so cap ham so lupng giac.............................................................. 174

5.8. Khao sat so cap ham so lupng giac ngupc................................................. 176

5.9. D o thj cua mpt so ham so chira dau gia trj tuyet doi................................. 179

5.10. Cac bai toan ve tiep tuyen............................................................................... 179

Bai tap.............................................................................................................................. 186

C huong 6 . BAT DANG T H U C 193

6.1. D^i cuong ve bat dang thirc.............................................................................. 193

6.2. Tinh chat cua bat dang thirc.............................................................................. 193

6.3. Mpt s p bat d5ng thirc thuang g$p.................................................................... 194

6.4. Chirng minh bat dang thirc............................................................................... 195

6.5. Cac bai tpan cue tri ham s p .............................................................................. 210

Bai tap............................................................................................................................ 220

Chircmg 7. PHLTONG TRINH , HE PH U O N G TRINH VA BAT

PHIJONG T R IN H ..................................................................................................... 226

7.1. D^i cucmg ve phucmg trinh, h$, tuyen phuong trinh va bat phuomg trinh.. 226

7.2. Phuong trinh, bat phuang trinh huu ti mpt an......................................... 234

4

7.3. Phuong trinh va he phuong trinh huru ti hai an............................................ 247

7.4 Bat phucrng trinh huu ti hai an......................................................................... 254

7.5. Phuong trinh va bat phuong trinh vo ti.......................................................... 255

7.6. Phuong trinh va bat phuong trinh mu, logarit............................................. 258

7.7. Phuong trinh va bat phuong trinh lupng giac.............................................. 262

Bai t?p.............................................................................................................................. 268

C huong 8 . GIAI BAI TOAN VA N H lfN G PIIEP SUY LIIAN

TR O N G GIAI T O A N ............................................................................................... 280

8 . L Cach giai mot bai toan....................................................................................... 280

8.2. Mot so phuong phap suy luan trong giai toan............................................. 284

Bai tap.............................................................................................................................. 294

TAl LIEU TIIAM KHAO......................................................................................... 299

5

L O IN O ID A U

Cuon giao trinh “Dai so so cap” dupe bien soan lain tai lieu hoc tap va

tham khao cho sinh vien nganh Sir pham Toan bac dai hoc. Theo chung toi,

nhung noi dung ma giao trinh de cap se phii hop nhat neu dupe giang day cho

sinh vien cuoi khoa sau khi sinh vien da tich liiy mot khoi lupng kha day du

nhung kien thuc chuyen mon nghiep vu can thiet thuoc nhieu ITnh vuc Toan

hoc. Viec hoc tap va nghien cuu ve dai so so cap cua sinh vien Su pham Toan

can phai chii trong den khau khai thac bai toan sau khi hoan thanh loi giai de

thay ro moi lien he giua bai toan da cho voi nhieu bai toan khae, hoan toan

khong chi dung lai a luyen giai cac dang toan - cong viec cua mot hoc sinh

thuan tuy. Hon the nua, chung toi ki vong ngircri hpc can nghien cuu moi chu

diem mot cach sang tao trong moi quan he mat thiet den mot loat npi dung

Toan hpc dupe trang bi p bac dai hpc nhu: Dai sp tuyen tinh, Dai so dai cupng,

Li thuyet so va cp so so hpc, Da thirc va phan thirc, Giai tich mot bien, Hinh

hpc,... nham dam bao tinh h? thong lien mon, dong thoi thay ro y nghTa thiet

thuc va lau dai cua chuong trinh Tpan bac dai hpc trpng dap tap giap vien Tpan

p pho thpng.

Giao trinh dupe cau true thanh tarn chuong. Tu Chuong 1 den Chuong 7,

chiing toi trinh bay ve cac chu de bao phii gan sat chuong trinh Dai so d pho

thong, bao gom: Tap hop so; Da thirc va phan thirc; Ham so va do thi; Bat dang

thuc; Phuong trinh; Bat phuong trinh. Chuong cuoi ciing dupe xem nhu mpt

phan tham khao them voi noi dung khai quat ve cac bupc thong thuong de giai

mot bai toan, cac huong thuong gap khi tien hanh khai thac mot bai toan sau

cong doan hoan thanh loi giai, mpt so phep suy luan quen thuoc trong giai toan.

Voi nhung ai da co nen tang co so ve Dai so so cap khi tiep can tai lieu nay

hoan toan co the dpc truoc Chuong 8 truoc khi nghien ciru cac chuong con lai.

Hien tai co nhieu tai lieu giao trinh ve Dai so so cap va nhirng van de lien quan,

chiing toi tham khao mot cach chon loc voi mong muon dua ra cach trinh bay

theo quan diem rieng doi voi mot ITnh vuc Toan so cap quen thuoc nhung ciing

rat phong phii va da dang. Khi trinh bay ve kien thirc, mot so noi dung Toan

hpc hien dai dupe chiing toi long ghep nham muc dich de nguoi dpc thay dupe

sir thong nhat va gan ket, bai qua trinh hinh thanh va pliat trien mot moil khoa

hoc dien ra lien tuc trong moi quan he mat thiet voi nhieu ITnh vuc khac, viec

phan chia ranh gioi So cap va Cao cap, Co dien va Hien dai nhieu luc chi mang

tinh tuang doi

Chung toi mong rang giao trinh la tai lieu hoc tap va tham khao huu ich

cho sinh vien theo hpc nganh Su pham Toan. Chung toi xin trail trong cam on

cac tac gia trong danh muc tai lieu tham khao ma chung toi da su dung khi bien

soan cuon sacli nay. Tri tue va kinh nghiem cua nhung nguoi di truoc la tien de

vung chac cho nhung nguoi di sau ke thira hpc tap. Qua day, chung toi xin bay

to long biet on chan thanh toi GS. TS. Bui Van Nglii va PGS TS. Vu Duong

Thuy - Truong Dai hpc Su pham Ha Npi da dpc ban thao va cho chung toi

nhieu y kien quy bau. Cuoi cung, chung toi xin chan thanh cam on Ban Giam

hieu, cac ban dong nghiep va cac can bo thuoc Phong Khoa hpc va Cong nghe -

Truong Dai hpc Hung Vupng (Phu Thp), Ban bien tap Nha Xuat ban Dai hpc

Thai Nguyen da giup do va tao dieu kien thuan loi cho giao trinh dupe hoan

thien va xuat ban. Lan dau tien bien soan nen giao trinh khong tranh khoi con

nhung thieu sot, rat mong don nhan dupe su gop y cua ban dpc.

Phii Thp, iif’ay OH Ihaiig 02 nam 2017

Nhoin tac gia

7

C hinm g 1

C A C T A P H O P SO

Chuong nay trinh bay ve each xay dung, cac tinh chat, quan he thir tu tren

cac tap hop so quen thuoc: tap so tu nhien N, tap so nguyen Z, tap so huu ti

Q, tap so thuc R, tap so phirc € va moi quan he giua chung.

1.1. So tir nhien

1.1. ]. Khai nient so tu nhien

Truoc het chung ta thira nhan: moi tap hop deu co mot han so, ban so cua

tap A dupe ki hi£u la \A\ hay Card (A). Ngoai ra cung quy uoc neu hai tap

A, B la tucmg duxmg (nghTa la co song anh tu tap nay den tap kia) thi co cung

ban so: |/l| = |/?|.

Mot tap hop duoc goi la him han neu no khong tuong duong voi bat ki

mpt bp phan thuc su cua np Ban so cua mot tap hop huu han dupe goi la mot

so hr nhien. Ki hieu tap tat ca cac so tu nhien la N. Qua day chung ta thay ban

so la khai niem mo rong cua “so lupng”.

D u6 i tu tuong tien de hoa, nam 1891 nha Toan hpc Italia la Peano

(Giuseppe Peano, 1858 - 1932) da dua ra h? tien de cho tap so tv nhien. Co the

noi phat minh cua Peano da lot ta dupe ban chat va cau true vpn co cua so tu

nhien, theo do sp tu nhien dupe cpi la mpt doi tinmg ca ban va giua cac so tu

nhien co san moi quan he ar han dau tien khong dinh nghTa Trong mo ta so tu

nhien theo ban so, su thira nhan ban so la cai von co gan lien voi moi tap hop

cung da chira dung dac tinh “khpng djnh nghTa dupe” cua doi tuong co ban.

He tien de Peano

Tnroc het chiing ta thira nhan rang co mpt tap hpp N ma mpi phan tir cua no

duoc goi la mpt so tu nhien. Giua cac so ty nhien “san co” mot quan he ke sau

thoa man cac tien de sau:

(i) Co mot so tu nhien khong ke sau bat ki so tu nhien nao khac (ki hieu la 0).

(ii) Moi so tu nhien luon ton tai mot va chi mot so ke sau

(iii) Moi so tir nhien khac 0 la so ke sau cua duy nhat mot so tir nhien khac.

(iv) Moi tap con A cua N sao cho A chira so 0 va neu A clriiu so tir nhien

n I hi A cung chira so ke sau cua n, thi A = N.

Trong phan nay, chung ta sir dung "mo hinh ban so" de trinh bay cac tinh

chat cua so ty nhien.

1.1.2. Quan he thir tic

Cho hai so tir nhien a,b. Gia sir A,B la hai tap hop thoa man

a = \A\, b = |/?|. Neu co mot don anh tir A den B thi ta noi a nho han hay bang

b va viet: a < b. Tap so tir nhien la N cung voi quan h? " <" noi tren la mpt lap

sap thir lu. Theo Cantor, luon co mpt don anh tir A den B hoac tir B den A va

neu co ca hai dieu nay thi A va B tuong duong voi nhau. Do do quan he thir tu

tren N la loan phan. Hon the nua, nguoi ta con chirng minh dupe tap so tv nhien

N la lap sap thir tu to! (mpi tap con khac rang luon co phan tir nho nhat).

1.1.3. Cac phep toan

a) Phep cong: Cho hai so tu nhien a,b. Gia sir A,B la hai tap hpp huu han

th6 a man a = \A\, h = |/j|, A f]B = O. T agpi |^U /^| la long cua hai s p tu nhien

a, b. Ki hieu la a + b.

b) Phep nhan: Cho hai so tu nhien a,b. Gia sir A,B la hai tap hop hiru han

thoa man a = |/<|, A = |/}|. Ta gpi |/lx tf| la tich cua hai s p tu nhien a, b. Ki

hieu ah.

c) Phep tru: Cho hai so tu nhien a,b. Neu co so tu nhien c sao cho a + c = b

thi ta noi c la hieu cua b va a. Ki hi?u c = b -a .

Dinh li 1.1.1. (i) Phep cong va phep nhan co tinh chat giao hoan, nghTa la

a + b = b + a, a b -b a v&i moi a, b e N.

(ii) Phep cong va phep nhan co tinh chat ket hcrp, nghTa la

(a + b) + c = a + (b + c), (ab)c = a(hc)

v&i moi a, b, c e N

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(iii) I’hep nhan co tinh chat phan phoi dai vai phep cong, nghTa Id:

a(b + c) = ab + ac vai moi a, b, c e N.

(iv) (N ,+ ), (N *,.)(vai N *= N \ {0 } ) la cac vi nhom giao hoan thoa man Inal

gian wrc, nghTa Id:

a + c = b + c ^ > a ~ h Va, b, c e N ,

ac = be => a = b V a, b e N, Vc e N *.

(v) I’hep cong va phep nhan co tinh chat bao toan quan he thir tu, nghTa Id

a < b => a + c < b + c vd a < b => ac < be V a, b, c e N.

Han nun, ta con co:

a< b= > a+ c< b + c V a, ft, c e N vd a < b => ac < be \/ a, b, c e N*.

(vi) Phep nhan co tinh chat phan phoi doi v&i phep trie, nghTa Id:

a(b - c) = a b -a c v&i moi a, b, c e N, ft > c.

d) Phep chia

Cho hai so tu nhien a,ft,ft / 0. Neu co so tu nhien c sao cho a = be thi

ta noi a chia het cho b (ki hieu a:b) hay a la hoi so cua b hoac b la ui'rc sii

cua a (ki hi£u b \ a), va gQi c la thuong cua a va b. Ki hi?u a :b=c.

Dinh Ii 1.1.2. Cho a ,b e N v&i ft * 0 .Khi do luon Ion lai duy nhat mot cap so

tu nhien {q, r) sao cho a = bq + r, 0 <r <b.

N guoi ta lan lupt goi q vd r la thuong va so d u cua phep chia a cho b.

c) Luy thira

C h o hai so ty nhien a, n (n y- 0). D at a" - a a ■ a v a d p c la a lu y thira

n

n. So a dupe goi la ca so, n goi la s p mu cua luy thira. Neu a * 0, ta quy uoc

a°= 1

Djnh Ii 1.1.3. Cho a, b, m, n e N. Ta co cac khang dinh sau:

(i) a V ” = a mf";

(ii) a" : a™ = a" m (n > m),

(iii) (a")m =a"m-

(iv) (a : b)n =a" :b" v&i a: ft.

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1.2. So nguyen

/. 2. /. Khai nicm vanh so nguyen

Djnh Ii 1.2.1. Ton lai mol mien nguyen Z vd mpt don anh f : N —> Z sao cho

(i) f vita la mpt don can nua nhom cong, vita la mot don can nua nhom nhan.

(ii) Cac phan tic cua Z co dang f(a)-f(b).

(iii) Cap (7L, f ) xdc dinh nhu hen la duy nhat sai khac mpt dang cdu, nghTa la

neti cap ( l ',g ) v&i I' Id mpt vanh vd g : N —> P Id mol dan anh, sao cho g

vim Id mot don cdu nua nhom cong, vim Id don cdu mm nhom nhan, vd moi

phan lit cua P deu cd dang g(a) - g(/>), thi Ion tai mpt dang cdu <p:Z—>P

sao cho (p f = g.

Djnh nghTa 1.2.1. Vanh Z xac djnh nhu tren duoc goi la vanh cac so nguyen.

Djnh Ii 1.2.2. Vanh cac so nguyen I Id vanh cut lieu, theo quan he bao ham,

chira tap cac so hr nhien N nhu- mol nua nhom cong vd niia nhom nhan. Ddo

lai, moi vanh cu t lieu chiiu lap cac so tu nhien N nhu mpt nua nhom cong vd

nim nhom nhan deu trung v&i Z (sai khac mpt dang cdu vanh).

Tiep theo, chung ta se nghien cuu ve tinh sap thir tu tren vanh so nguyen.

Truoc het xin nhac lai mot so khai niem va ket qua ve vanh sap thir tu

a) V anh sap thu tu

Djnh nghTa 1.2.2. Vanh giao hoan A co don vi I ^ 0 duoc gpi la mot vanh sap

thir tu neu A dupe trang bj mpt quan he thir tu toan phan > thoa man dupe hai

dieu kien sau:

( i ) r -^ k e p t h e p x i z >y i z voi mpi x,y,zeA.

(ii) V 6 i mpi x,ye A, ta c o x > 0 va_y > 0 keo th eox^ > 0.

Vanh A sap thu tu dupe gpi la mot vanh sap thir tu Archimede neu voi mpi

x ,y e A va x >0 deu ton tai mot so tu nhien n de nx > y. Mot bo phan M cua

vanh sap thir tu A duoc gpi la bi chan tren (chan dux'ri) neu ton tai mpt phan tir

a e A sao cho a > x (x > a) voi mpi x e M . Mot bo phan cua vanh sap thir tu

dupe gpi la hi chan neu no vira bj chan tren, vira bi chan duoi.

Voi A la mpt vanh sap thir tu, thi do x + (-x) = 0 nen voi moi x ta co hoac x

> 0 hoac —x > 0 . Lai do 1 = 1.1 = (-1 ).( - 1 ) va x2 = x.x = (-x)( -x) nen 1 > 0 va

x2 > 0. Bai 1 > 0 nen n 1 > 1 > 0 vai moi so tir nhien n > 0. Vay vanh sap thir tir

la mot vanh co dac so 0 . Voi moi x £ 0, ta luon co hoac x > 0 hoac x < 0. Cac

phan tir x > 0 dupe goi la cac phan tir duan#, cac phan tu y < 0 dupe gpi la cac

phan tu am. Ki hieu A' va A la tap cac phan tu duang va am tuong irng cua

A. Tir cac lap luan tren ta lap tire rut ra A' r\A = <t> va A = A' u / T u { 0 } .

Ngoai ra, xuat phat tir x(-y) = -xy, nen tir xy > 0 va x > 0 thi suy ra^ > 0.

Trong vanh sap thir tu A, nguoi ta dua ra khai mem trj tuyet doi cua mot

phan tir nhu sau:

Djnh nghia 1.2.3. Tri tuyet doi cua mot phan tir a e A , ki hieu |a|, dupe xac

djnh boi

a khi a >0

[-cr khi a < 0

Tri tuyet doi co cac tinh chat sau.

Menh de 1.2.1. Cho a, b, c\ d Id cac phan tie cua mol trurmg sap thir tu vai c f 0,

d > 0. Khi do ta co:

(i) |a/>| r

ki

(i i) \a + b\ < \a\ + 1/>|, ||a| - |ft|| < \a - b\ < |a| + |ft|.

(iii) |a| < d khi vd chi khi -d <a <d.

b) Quan he thu- tu' trong Z

Djnh nghia 1.2.4. Trong Z ta xac dinh cac quan he tir tu > nhu sau: voi mpi

x, y e Z, x > y neu va chi neu x - y e N .

Quan he thir tu trong Z co cac tinh chat sau.

M enh de 1.2.2. Quan he > xac dinh nhu tren la mot quan he thu tu toan phan

trong Z .

Djnh Ii 1.2.3. Vanh cac so nguyen Z vai quan he thu tu da ditrrc xac dmh, la

mot vanh sap thir tu Archimede.

M enh de 1.2.3. Neu x > y thi x > y +1.

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Dinh Ii 1.2.4. Mpi bp phan cua vanh cac so nguyen Z bi chan tren, deu co

phan tu Ion nhat. Mpi bp phan cua vanh cac so nguyen Z bi chan dut'ri, deu co

phan lu nho nhat.

1.2.2. Li thuyet chia het tren vanh so nguyen

a) Quan he chia het

Djnh nghia 1.2.5. So nguyen a dupe gpi la chia het cho mpt so nguyen b, hay b

chia het cho a neu ton tai mpt so nguyen c sao cho a = be. Khi a chia het cho b

ta viet a.b hoac b \ a va b dupe gpi la ux'rc cua a, con a dupe gpi la bpi cua b.

C hu y 1.2.1. So nguyen a chia het cho 0 khi va chi khi a = 0. Do do bpi cua

so 0 chi la 0. Tuy nhien tap cac uoc cua 0 lai la toan bo Z.

Cac tinh chat ca ban.

(i) 1 | a voi mpi a e Z va a | a voi mpi a e Z.

(ii) Neu a \ b va b \ c thi a \ c.

(iii) Neu b * 0 va a | b thi \a\ < |A|.

n

(iv) Neu a \ b, thi a | xt voi mpi x, e Z.

I I

(v) Neu a | b va b | a thi a = b hoac a = -b

(vi) Quan he chia het trong Z co tinh phan xa, bac cau, nhung khong co

tinh doi xirng.

(vii) Quan he chia het trong Z co tinh phan doi xirng.

b) Phep chia voi du1

Djnh Ii 1.2.5. Cho a. b g Z. b ^ 0 Khi do ton tai duy nhat nipt <ijp so nguyen

q, r vai 0 < r < |ft| de a = bq + r.

1.2.3. U'&c chung It'm nhat - Hoi chung nho nhat

a) U oc chung Ion nhat

Djnh nghia 1.2.6. Cho au ...,a n,d e 7L Khi do:

(i) d dupe gpi la mot ut'rc chung cua cac so a{,...,a n neu d \ a: (/ = !,...,« ).

(ii) d dupe gpi la ux'rc chung lan nhat cua al,...,a n neu d chia het cho mpi irdc

chung cua

13

Chu y 1.2.2. Cho a ,,...,a n e Z . Khi do:

(i) Neu a ,,...,a n khong dong thai bang 0 thi tap cac irac chung cua la

huu han va khac rong Trong truang hap nay al,...,a n co hai uac chung lan

nhat la hai so doi nhau. Ki hieu (a ,,.,.,an) la so duang trong hai so nay. De

thay rang (a x,...,a n) la so Ian nhat trong tat ca cac uac chung cua at,...,a n .

(ii)N eu (ax,...,a n) = 1 thi a ,, . . . , an dupe gpi la nguyen Id ciing nhau.

(iii) Cac so a{,...,a n dupe gpi la nguyen Id sanh doi, hay doi mot nguyen Id

ciing nhau neu (a, , ) = 1 vai mpi i , j = \,...,n , / * j.

(iv) Neu a, = a2 = -- = an = 0 thi ( a ,,...,a „ ) = 0 .

(v) (0 , ax,...,a„) = ( a „ . . . , a j .

(vi) ( l ,a „ ...,o J = ( - l , au ...,a n) = l

(vii) (a1, . . . )a„) = ((a l,...,fl„_1),a„).

Tinh chat nay chi ra cach tim upc chung 16n nhat cua nhieu sp dupe quy ve

viec tim uac chung lan nhat cua 2 so.

(viii) (ka],...,k a n) = \k\(al,...,a ll)v6\ mpi k e Z.

(ix) Cho a ,b ,c ,d e Z sao cho a = be + d. Khi do ta luon co: (a,ft) = (ft,t/).

Thuat toan Euclid: Gia su a va b la hai so nguyen duang vai a > b va dat

r„ = a, rx = ft. Bang cach ap dung lien tiep thuat toan chia, ta dupe:

ro = W + ri

r\ = r2q2 + r,

r„-i = rn xqn , + r

rn-\ =

vai rt >r2 > ■■■ > rn > 0 . Cuoi cung, so 0 se xuat hien trong day phep chia

lien tiep, vi day cac so du ft = /; > r 2 > •••> r„ > 0 khong the chira qua ft

so. Hon nua tir (ix) ta suy ra

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