NCERT पाठ्यपुस्तक पाठ 1 - वास्तविक संख्याएँ, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 1 - वास्तविक संख्याएँ, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


okLrfod la[;k,¡ 1
1
1.1 Hkwfedk
d{kk 9 esa] vkius okLrfod la[;kvksa dh [kkst izkjaHk dh vkSj bl izfØ;k ls vkidks
vifjes; la[;kvksa dks tkuus dk volj feykA bl vè;k; esa] ge okLrfod la[;kvksa
osQ ckjs esa viuh ppkZ tkjh j[ksaxsA ;g ppkZ ge vuqPNsn 1-2 rFkk 1-3 esa /ukRed iw.kk±dksa
osQ nks vfr egRoiw.kZ xq.kksa ls izkjaHk djsaxsA ;s xq.k gSa% ;wfDyM foHkktu ,YxksfjFe (dyu
fof/) (Euclid’s division algorithm) vkSj vadxf.kr dh vk/kjHkwr izes; (Fundamental
Theorem of Arithmetic) A
tSlk fd uke ls fofnr gksrk gS] ;wfDyM foHkktu ,YxksfjFe iw.kk±dksa dh foHkkT;rk
ls fdlh :i esa lacaf/r gSA lk/kj.k Hkk"kk esa dgk tk,] rks ,YxksfjFe osQ vuqlkj] ,d
/ukRed iw.kk±d a dks fdlh vU; /ukRed iw.kk±d b ls bl izdkj foHkkftr fd;k tk
ldrk gS fd 'ks"kiQy  r  izkIr gks] tks b ls NksVk (de) gSA vki esa ls vf/drj yksx 'kk;n
bls lkekU; yach foHkktu izfØ;k (long division process) osQ :i esa tkurs gSaA ;|fi ;g
ifj.kke dgus vkSj le>us esa cgqr ljy gS] ijarq iw.kk±dksa dh foHkkT;rk osQ xq.kksa ls lacafèkr
blosQ vusd vuqiz;ksx gSaA ge buesa ls oqQN ij izdk'k Mkysaxs rFkk eq[;r% bldk iz;ksx
nks /ukRed iw.kk±dksa dk egÙke lekiorZd (HCF) ifjdfyr djus esa djsaxsA
nwljh vksj] vadxf.kr dh vk/kjHkwr izes; dk laca/ /ukRed iw.kk±dksa osQ xq.ku ls
gSA vki igys ls gh tkurs gSa fd izR;sd HkkT; la[;k (Composite number) dks ,d
vf}rh; :i ls vHkkT; la[;kvksa (prime numbers) osQ xq.kuiQy osQ :i esa O;Dr fd;k
tk ldrk gSA ;gh egRoiw.kZ rF; vadxf.kr dh vk/kjHkwr izes; gSA iqu%] ;g ifj.kke
dgus vkSj le>us esa cgqr ljy gS] ijarq blosQ xf.kr osQ {ks=k esa cgqr O;kid vkSj lkFkZd
vuqiz;ksx gSaA ;gk¡] ge vadxf.kr dh vk/kjHkwr izes; osQ nks eq[; vuqiz;ksx ns[ksaxsA ,d
okLrfod la[;k,¡
2018-19
Page 2


okLrfod la[;k,¡ 1
1
1.1 Hkwfedk
d{kk 9 esa] vkius okLrfod la[;kvksa dh [kkst izkjaHk dh vkSj bl izfØ;k ls vkidks
vifjes; la[;kvksa dks tkuus dk volj feykA bl vè;k; esa] ge okLrfod la[;kvksa
osQ ckjs esa viuh ppkZ tkjh j[ksaxsA ;g ppkZ ge vuqPNsn 1-2 rFkk 1-3 esa /ukRed iw.kk±dksa
osQ nks vfr egRoiw.kZ xq.kksa ls izkjaHk djsaxsA ;s xq.k gSa% ;wfDyM foHkktu ,YxksfjFe (dyu
fof/) (Euclid’s division algorithm) vkSj vadxf.kr dh vk/kjHkwr izes; (Fundamental
Theorem of Arithmetic) A
tSlk fd uke ls fofnr gksrk gS] ;wfDyM foHkktu ,YxksfjFe iw.kk±dksa dh foHkkT;rk
ls fdlh :i esa lacaf/r gSA lk/kj.k Hkk"kk esa dgk tk,] rks ,YxksfjFe osQ vuqlkj] ,d
/ukRed iw.kk±d a dks fdlh vU; /ukRed iw.kk±d b ls bl izdkj foHkkftr fd;k tk
ldrk gS fd 'ks"kiQy  r  izkIr gks] tks b ls NksVk (de) gSA vki esa ls vf/drj yksx 'kk;n
bls lkekU; yach foHkktu izfØ;k (long division process) osQ :i esa tkurs gSaA ;|fi ;g
ifj.kke dgus vkSj le>us esa cgqr ljy gS] ijarq iw.kk±dksa dh foHkkT;rk osQ xq.kksa ls lacafèkr
blosQ vusd vuqiz;ksx gSaA ge buesa ls oqQN ij izdk'k Mkysaxs rFkk eq[;r% bldk iz;ksx
nks /ukRed iw.kk±dksa dk egÙke lekiorZd (HCF) ifjdfyr djus esa djsaxsA
nwljh vksj] vadxf.kr dh vk/kjHkwr izes; dk laca/ /ukRed iw.kk±dksa osQ xq.ku ls
gSA vki igys ls gh tkurs gSa fd izR;sd HkkT; la[;k (Composite number) dks ,d
vf}rh; :i ls vHkkT; la[;kvksa (prime numbers) osQ xq.kuiQy osQ :i esa O;Dr fd;k
tk ldrk gSA ;gh egRoiw.kZ rF; vadxf.kr dh vk/kjHkwr izes; gSA iqu%] ;g ifj.kke
dgus vkSj le>us esa cgqr ljy gS] ijarq blosQ xf.kr osQ {ks=k esa cgqr O;kid vkSj lkFkZd
vuqiz;ksx gSaA ;gk¡] ge vadxf.kr dh vk/kjHkwr izes; osQ nks eq[; vuqiz;ksx ns[ksaxsA ,d
okLrfod la[;k,¡
2018-19
2 xf.kr
rks ge bldk iz;ksx d{kk IX esa vè;;u dh xbZ oqQN la[;kvksa] tSls 2, 3 vkSj 
5
vkfn dh vifjes;rk fl¼ djus esa djsaxsA nwljs] ge bldk iz;ksx ;g [kkstus esa djsaxs fd
fdlh ifjes; la[;k] eku yhft, ( 0)
p
q
q
? , dk n'keyo izlkj dc lkar (terminating)
gksrk gS rFkk dc vlkar vkorhZ (non-terminating repeating) gksrk gSA ,slk ge 
p
q
osQ gj
q osQ vHkkT; xq.ku[kaMu dks ns[kdj Kkr djrs gSaA vki ns[ksaxs fd q osQ vHkkT; xq.ku[kaMu
ls 
p
q
osQ n'keyo izlkj dh izo`Qfr dk iw.kZr;k irk yx tk,xkA
vr%] vkb, viuh [kkst izkjaHk djsaA
1.2 ;wfDyM foHkktu izesf;dk
fuEufyf[kr yksd igsyh* ij fopkj dhft,%
,d foozsQrk lM+d ij pyrs gq, vaMs csp jgk FkkA ,d vkylh O;fDr] ftlosQ ikl
dksbZ dke ugha Fkk] us ml foozsQrk ls oko~Q&;q¼ izkjaHk dj fn;kA blls ckr vkxs c<+ xbZ
vkSj mlus vaMksa dh Vksdjh dks Nhu dj lM+d ij fxjk fn;kA vaMs VwV x,A fooszQrk us
iapk;r ls dgk fd ml O;fDr ls VwVs gq, vaMksa dk ewY; nsus dks dgsA iapk;r us fooszQrk
ls iwNk fd fdrus vaMs VwVs FksA mlus fuEufyf[kr mÙkj fn;k%
nks&nks fxuus ij ,d cpsxk_
rhu&rhu fxuus ij nks cpsaxs_
pkj&pkj fxuus ij rhu cpsaxs_
ik¡p&ik¡p fxuus ij pkj cpsaxs_
N%&N% fxuus ij ik¡p cpsaxs_
lkr&lkr fxuus ij oqQN ugha cpsxk_
esjh Vksdjh eas 150 ls vf/d vaMs ugha vk ldrsA
vr%] fdrus vaMs Fks\ vkb, bl igsyh dks gy djus dk iz;Ru djsaA eku yhft,
vaMksa dh la[;k a gSA rc mYVs Øe ls dk;Z djrs gq,] ge ns[krs gSa fd a la[;k 150 ls
NksVh gS ;k mlosQ cjkcj gSA
;fn lkr&lkr fxusa] rks oqQN ugha cpsxkA ;g a = 7p + 0 osQ :i esa ifjo£rr gks tkrk
gS] tgk¡ p dksbZ izko`Qr la[;k gSA
* ;g ^U;wesjslh dkmaV~l* (ys[kdx.k ,- jkeiky vkSj vU;) esa nh igsyh dk ,d ifjo£rr :i gSA
2018-19
Page 3


okLrfod la[;k,¡ 1
1
1.1 Hkwfedk
d{kk 9 esa] vkius okLrfod la[;kvksa dh [kkst izkjaHk dh vkSj bl izfØ;k ls vkidks
vifjes; la[;kvksa dks tkuus dk volj feykA bl vè;k; esa] ge okLrfod la[;kvksa
osQ ckjs esa viuh ppkZ tkjh j[ksaxsA ;g ppkZ ge vuqPNsn 1-2 rFkk 1-3 esa /ukRed iw.kk±dksa
osQ nks vfr egRoiw.kZ xq.kksa ls izkjaHk djsaxsA ;s xq.k gSa% ;wfDyM foHkktu ,YxksfjFe (dyu
fof/) (Euclid’s division algorithm) vkSj vadxf.kr dh vk/kjHkwr izes; (Fundamental
Theorem of Arithmetic) A
tSlk fd uke ls fofnr gksrk gS] ;wfDyM foHkktu ,YxksfjFe iw.kk±dksa dh foHkkT;rk
ls fdlh :i esa lacaf/r gSA lk/kj.k Hkk"kk esa dgk tk,] rks ,YxksfjFe osQ vuqlkj] ,d
/ukRed iw.kk±d a dks fdlh vU; /ukRed iw.kk±d b ls bl izdkj foHkkftr fd;k tk
ldrk gS fd 'ks"kiQy  r  izkIr gks] tks b ls NksVk (de) gSA vki esa ls vf/drj yksx 'kk;n
bls lkekU; yach foHkktu izfØ;k (long division process) osQ :i esa tkurs gSaA ;|fi ;g
ifj.kke dgus vkSj le>us esa cgqr ljy gS] ijarq iw.kk±dksa dh foHkkT;rk osQ xq.kksa ls lacafèkr
blosQ vusd vuqiz;ksx gSaA ge buesa ls oqQN ij izdk'k Mkysaxs rFkk eq[;r% bldk iz;ksx
nks /ukRed iw.kk±dksa dk egÙke lekiorZd (HCF) ifjdfyr djus esa djsaxsA
nwljh vksj] vadxf.kr dh vk/kjHkwr izes; dk laca/ /ukRed iw.kk±dksa osQ xq.ku ls
gSA vki igys ls gh tkurs gSa fd izR;sd HkkT; la[;k (Composite number) dks ,d
vf}rh; :i ls vHkkT; la[;kvksa (prime numbers) osQ xq.kuiQy osQ :i esa O;Dr fd;k
tk ldrk gSA ;gh egRoiw.kZ rF; vadxf.kr dh vk/kjHkwr izes; gSA iqu%] ;g ifj.kke
dgus vkSj le>us esa cgqr ljy gS] ijarq blosQ xf.kr osQ {ks=k esa cgqr O;kid vkSj lkFkZd
vuqiz;ksx gSaA ;gk¡] ge vadxf.kr dh vk/kjHkwr izes; osQ nks eq[; vuqiz;ksx ns[ksaxsA ,d
okLrfod la[;k,¡
2018-19
2 xf.kr
rks ge bldk iz;ksx d{kk IX esa vè;;u dh xbZ oqQN la[;kvksa] tSls 2, 3 vkSj 
5
vkfn dh vifjes;rk fl¼ djus esa djsaxsA nwljs] ge bldk iz;ksx ;g [kkstus esa djsaxs fd
fdlh ifjes; la[;k] eku yhft, ( 0)
p
q
q
? , dk n'keyo izlkj dc lkar (terminating)
gksrk gS rFkk dc vlkar vkorhZ (non-terminating repeating) gksrk gSA ,slk ge 
p
q
osQ gj
q osQ vHkkT; xq.ku[kaMu dks ns[kdj Kkr djrs gSaA vki ns[ksaxs fd q osQ vHkkT; xq.ku[kaMu
ls 
p
q
osQ n'keyo izlkj dh izo`Qfr dk iw.kZr;k irk yx tk,xkA
vr%] vkb, viuh [kkst izkjaHk djsaA
1.2 ;wfDyM foHkktu izesf;dk
fuEufyf[kr yksd igsyh* ij fopkj dhft,%
,d foozsQrk lM+d ij pyrs gq, vaMs csp jgk FkkA ,d vkylh O;fDr] ftlosQ ikl
dksbZ dke ugha Fkk] us ml foozsQrk ls oko~Q&;q¼ izkjaHk dj fn;kA blls ckr vkxs c<+ xbZ
vkSj mlus vaMksa dh Vksdjh dks Nhu dj lM+d ij fxjk fn;kA vaMs VwV x,A fooszQrk us
iapk;r ls dgk fd ml O;fDr ls VwVs gq, vaMksa dk ewY; nsus dks dgsA iapk;r us fooszQrk
ls iwNk fd fdrus vaMs VwVs FksA mlus fuEufyf[kr mÙkj fn;k%
nks&nks fxuus ij ,d cpsxk_
rhu&rhu fxuus ij nks cpsaxs_
pkj&pkj fxuus ij rhu cpsaxs_
ik¡p&ik¡p fxuus ij pkj cpsaxs_
N%&N% fxuus ij ik¡p cpsaxs_
lkr&lkr fxuus ij oqQN ugha cpsxk_
esjh Vksdjh eas 150 ls vf/d vaMs ugha vk ldrsA
vr%] fdrus vaMs Fks\ vkb, bl igsyh dks gy djus dk iz;Ru djsaA eku yhft,
vaMksa dh la[;k a gSA rc mYVs Øe ls dk;Z djrs gq,] ge ns[krs gSa fd a la[;k 150 ls
NksVh gS ;k mlosQ cjkcj gSA
;fn lkr&lkr fxusa] rks oqQN ugha cpsxkA ;g a = 7p + 0 osQ :i esa ifjo£rr gks tkrk
gS] tgk¡ p dksbZ izko`Qr la[;k gSA
* ;g ^U;wesjslh dkmaV~l* (ys[kdx.k ,- jkeiky vkSj vU;) esa nh igsyh dk ,d ifjo£rr :i gSA
2018-19
okLrfod la[;k,¡ 3
;fn N%&N% fxusa] rks 5 cpsaxsA ;g a = 6q + 5 osQ :i esa ifjo£rr gks tkrk gS] tgk¡
q dksbZ izko`Qr la[;k gSA
ik¡p&ik¡p fxuus ij] 4 cpsaxsA ;g a = 5s + 4 esa ifjo£rr gks tkrk gS] tgk¡ s dksbZ
izko`Qr la[;k gSA
pkj&pkj fxuus ij] 3 cpsaxsA ;g a = 4t + 3, esa ifjo£rr gks tkrk gS] tgk¡ t dksbZ
izko`Qr la[;k gSA
rhu&rhu fxuus ij 2 cpsaxsA ;g a = 3u + 2 esa ifjo£rr gks tkrk gS] tgk¡ u dksbZ
izko`Qr la[;k gSA
nks&nks fxuus ij] 1 cpsxkA ;g a = 2v + 1, esa ifjo£rr gks tkrk gS tgk¡ v dksbZ izko`Qr la[;k gSA
vFkkZr~] mijksDr izR;sd fLFkfr esa] gekjs ikl nks /ukRed iw.kk±d a vkSj b gSa (fy, x,
mnkgj.k esa b osQ eku Øe'k% 7, 6, 5, 4, 3 vkSj 2 gSa)A buesa a dks b ls Hkkx nsus ij 'ks"k r
cprk gS (mijksDr esa r osQ eku Øe'k% 0, 5, 4, 3, 2 vkSj 1 gSa) vFkkZr~] r Hkktd b ls NksVk
gSA tSls gh ge bl izdkj osQ lehdj.k fy[krs gSa] ge ;wfDyM foHkktu izesf;dk
(Euclid’s division lemma) dk iz;ksx dj jgs gSa] ftls izes; 1-1 esa fn;k tk jgk gSA
vc viuh igsyh ij okil vkus ij] D;k vki dksbZ ckr lksp dj crk ldrs gSa
fd bl igsyh dks oSQls gy djsaxs\ gk¡! vki 7 osQ ,sls xq.ktksa dks [kksft, tks mijksDr lHkh
izfrca/ksa dks larq"V djasA tk¡p vkSj Hkwy fof/ ls (LCM dk iz;ksx djosQ) vki Kkr dj
ldrs gSa fd vaMksa dh la[;k 119 FkhA
bl ckr dk vuqHko djus osQ fy, fd ;wfDyM foHkktu izesf;dk D;k gS] iw.kk±dksa
osQ fuEufyf[kr ;qXeksa ij fopkj dhft,%
(i) 17, 6 (ii) 5, 12 (iii) 20, 4
tSlk fd geus igsyh okys mnkgj.k esa fd;k Fkk] ;gk¡ Hkh ge izR;sd ;qXe osQ fy,
laca/ fy[k ldrs gSa tSlk fd uhps n'kkZ;k x;k gSA
(i) 17 = 6 × 2 + 5 (17 esa 6 nks ckj tkrk gS vkSj 'ks"k 5 cprk gS)
(ii) 5 = 12 × 0 + 5 (;g laca/ blfy, lgh gS] D;ksafd 12] 5 ls cM+k gS)
(iii) 20 = 4 × 5 + 0 (20 esa 4 ik¡p ckj tkrk gS vkSj oqQN 'ks"k ugha cprk)
vFkkZr~ /ukRed iw.kk±dksa  a vkSj b osQ izR;sd ;qXe osQ fy,] geus ,slh iw.kZ la[;k,¡  q
vkSj r Kkr dj pqosQ gSa fd
a = bq + r, 0 = r < b gSA
2018-19
Page 4


okLrfod la[;k,¡ 1
1
1.1 Hkwfedk
d{kk 9 esa] vkius okLrfod la[;kvksa dh [kkst izkjaHk dh vkSj bl izfØ;k ls vkidks
vifjes; la[;kvksa dks tkuus dk volj feykA bl vè;k; esa] ge okLrfod la[;kvksa
osQ ckjs esa viuh ppkZ tkjh j[ksaxsA ;g ppkZ ge vuqPNsn 1-2 rFkk 1-3 esa /ukRed iw.kk±dksa
osQ nks vfr egRoiw.kZ xq.kksa ls izkjaHk djsaxsA ;s xq.k gSa% ;wfDyM foHkktu ,YxksfjFe (dyu
fof/) (Euclid’s division algorithm) vkSj vadxf.kr dh vk/kjHkwr izes; (Fundamental
Theorem of Arithmetic) A
tSlk fd uke ls fofnr gksrk gS] ;wfDyM foHkktu ,YxksfjFe iw.kk±dksa dh foHkkT;rk
ls fdlh :i esa lacaf/r gSA lk/kj.k Hkk"kk esa dgk tk,] rks ,YxksfjFe osQ vuqlkj] ,d
/ukRed iw.kk±d a dks fdlh vU; /ukRed iw.kk±d b ls bl izdkj foHkkftr fd;k tk
ldrk gS fd 'ks"kiQy  r  izkIr gks] tks b ls NksVk (de) gSA vki esa ls vf/drj yksx 'kk;n
bls lkekU; yach foHkktu izfØ;k (long division process) osQ :i esa tkurs gSaA ;|fi ;g
ifj.kke dgus vkSj le>us esa cgqr ljy gS] ijarq iw.kk±dksa dh foHkkT;rk osQ xq.kksa ls lacafèkr
blosQ vusd vuqiz;ksx gSaA ge buesa ls oqQN ij izdk'k Mkysaxs rFkk eq[;r% bldk iz;ksx
nks /ukRed iw.kk±dksa dk egÙke lekiorZd (HCF) ifjdfyr djus esa djsaxsA
nwljh vksj] vadxf.kr dh vk/kjHkwr izes; dk laca/ /ukRed iw.kk±dksa osQ xq.ku ls
gSA vki igys ls gh tkurs gSa fd izR;sd HkkT; la[;k (Composite number) dks ,d
vf}rh; :i ls vHkkT; la[;kvksa (prime numbers) osQ xq.kuiQy osQ :i esa O;Dr fd;k
tk ldrk gSA ;gh egRoiw.kZ rF; vadxf.kr dh vk/kjHkwr izes; gSA iqu%] ;g ifj.kke
dgus vkSj le>us esa cgqr ljy gS] ijarq blosQ xf.kr osQ {ks=k esa cgqr O;kid vkSj lkFkZd
vuqiz;ksx gSaA ;gk¡] ge vadxf.kr dh vk/kjHkwr izes; osQ nks eq[; vuqiz;ksx ns[ksaxsA ,d
okLrfod la[;k,¡
2018-19
2 xf.kr
rks ge bldk iz;ksx d{kk IX esa vè;;u dh xbZ oqQN la[;kvksa] tSls 2, 3 vkSj 
5
vkfn dh vifjes;rk fl¼ djus esa djsaxsA nwljs] ge bldk iz;ksx ;g [kkstus esa djsaxs fd
fdlh ifjes; la[;k] eku yhft, ( 0)
p
q
q
? , dk n'keyo izlkj dc lkar (terminating)
gksrk gS rFkk dc vlkar vkorhZ (non-terminating repeating) gksrk gSA ,slk ge 
p
q
osQ gj
q osQ vHkkT; xq.ku[kaMu dks ns[kdj Kkr djrs gSaA vki ns[ksaxs fd q osQ vHkkT; xq.ku[kaMu
ls 
p
q
osQ n'keyo izlkj dh izo`Qfr dk iw.kZr;k irk yx tk,xkA
vr%] vkb, viuh [kkst izkjaHk djsaA
1.2 ;wfDyM foHkktu izesf;dk
fuEufyf[kr yksd igsyh* ij fopkj dhft,%
,d foozsQrk lM+d ij pyrs gq, vaMs csp jgk FkkA ,d vkylh O;fDr] ftlosQ ikl
dksbZ dke ugha Fkk] us ml foozsQrk ls oko~Q&;q¼ izkjaHk dj fn;kA blls ckr vkxs c<+ xbZ
vkSj mlus vaMksa dh Vksdjh dks Nhu dj lM+d ij fxjk fn;kA vaMs VwV x,A fooszQrk us
iapk;r ls dgk fd ml O;fDr ls VwVs gq, vaMksa dk ewY; nsus dks dgsA iapk;r us fooszQrk
ls iwNk fd fdrus vaMs VwVs FksA mlus fuEufyf[kr mÙkj fn;k%
nks&nks fxuus ij ,d cpsxk_
rhu&rhu fxuus ij nks cpsaxs_
pkj&pkj fxuus ij rhu cpsaxs_
ik¡p&ik¡p fxuus ij pkj cpsaxs_
N%&N% fxuus ij ik¡p cpsaxs_
lkr&lkr fxuus ij oqQN ugha cpsxk_
esjh Vksdjh eas 150 ls vf/d vaMs ugha vk ldrsA
vr%] fdrus vaMs Fks\ vkb, bl igsyh dks gy djus dk iz;Ru djsaA eku yhft,
vaMksa dh la[;k a gSA rc mYVs Øe ls dk;Z djrs gq,] ge ns[krs gSa fd a la[;k 150 ls
NksVh gS ;k mlosQ cjkcj gSA
;fn lkr&lkr fxusa] rks oqQN ugha cpsxkA ;g a = 7p + 0 osQ :i esa ifjo£rr gks tkrk
gS] tgk¡ p dksbZ izko`Qr la[;k gSA
* ;g ^U;wesjslh dkmaV~l* (ys[kdx.k ,- jkeiky vkSj vU;) esa nh igsyh dk ,d ifjo£rr :i gSA
2018-19
okLrfod la[;k,¡ 3
;fn N%&N% fxusa] rks 5 cpsaxsA ;g a = 6q + 5 osQ :i esa ifjo£rr gks tkrk gS] tgk¡
q dksbZ izko`Qr la[;k gSA
ik¡p&ik¡p fxuus ij] 4 cpsaxsA ;g a = 5s + 4 esa ifjo£rr gks tkrk gS] tgk¡ s dksbZ
izko`Qr la[;k gSA
pkj&pkj fxuus ij] 3 cpsaxsA ;g a = 4t + 3, esa ifjo£rr gks tkrk gS] tgk¡ t dksbZ
izko`Qr la[;k gSA
rhu&rhu fxuus ij 2 cpsaxsA ;g a = 3u + 2 esa ifjo£rr gks tkrk gS] tgk¡ u dksbZ
izko`Qr la[;k gSA
nks&nks fxuus ij] 1 cpsxkA ;g a = 2v + 1, esa ifjo£rr gks tkrk gS tgk¡ v dksbZ izko`Qr la[;k gSA
vFkkZr~] mijksDr izR;sd fLFkfr esa] gekjs ikl nks /ukRed iw.kk±d a vkSj b gSa (fy, x,
mnkgj.k esa b osQ eku Øe'k% 7, 6, 5, 4, 3 vkSj 2 gSa)A buesa a dks b ls Hkkx nsus ij 'ks"k r
cprk gS (mijksDr esa r osQ eku Øe'k% 0, 5, 4, 3, 2 vkSj 1 gSa) vFkkZr~] r Hkktd b ls NksVk
gSA tSls gh ge bl izdkj osQ lehdj.k fy[krs gSa] ge ;wfDyM foHkktu izesf;dk
(Euclid’s division lemma) dk iz;ksx dj jgs gSa] ftls izes; 1-1 esa fn;k tk jgk gSA
vc viuh igsyh ij okil vkus ij] D;k vki dksbZ ckr lksp dj crk ldrs gSa
fd bl igsyh dks oSQls gy djsaxs\ gk¡! vki 7 osQ ,sls xq.ktksa dks [kksft, tks mijksDr lHkh
izfrca/ksa dks larq"V djasA tk¡p vkSj Hkwy fof/ ls (LCM dk iz;ksx djosQ) vki Kkr dj
ldrs gSa fd vaMksa dh la[;k 119 FkhA
bl ckr dk vuqHko djus osQ fy, fd ;wfDyM foHkktu izesf;dk D;k gS] iw.kk±dksa
osQ fuEufyf[kr ;qXeksa ij fopkj dhft,%
(i) 17, 6 (ii) 5, 12 (iii) 20, 4
tSlk fd geus igsyh okys mnkgj.k esa fd;k Fkk] ;gk¡ Hkh ge izR;sd ;qXe osQ fy,
laca/ fy[k ldrs gSa tSlk fd uhps n'kkZ;k x;k gSA
(i) 17 = 6 × 2 + 5 (17 esa 6 nks ckj tkrk gS vkSj 'ks"k 5 cprk gS)
(ii) 5 = 12 × 0 + 5 (;g laca/ blfy, lgh gS] D;ksafd 12] 5 ls cM+k gS)
(iii) 20 = 4 × 5 + 0 (20 esa 4 ik¡p ckj tkrk gS vkSj oqQN 'ks"k ugha cprk)
vFkkZr~ /ukRed iw.kk±dksa  a vkSj b osQ izR;sd ;qXe osQ fy,] geus ,slh iw.kZ la[;k,¡  q
vkSj r Kkr dj pqosQ gSa fd
a = bq + r, 0 = r < b gSA
2018-19
4 xf.kr
è;ku nhft, fd  q  ;k  r 'kwU; Hkh gks ldrs gSaA
vc vki /ukRed iw.kk±dksa  a vkSj b osQ fuEufyf[kr ;qXeksa osQ fy, iw.kk±d q vkSj r
Kkr djus dk iz;Ru dhft,%
(i) 10, 3 (ii) 4, 19 (iii) 81, 3
D;k vki è;ku ns jgs gSa fd q vkSj r vf}rh; gSa\ ;s gh osQoy ,sls iw.kk±d gSa] tks
izfrca/ksa  a = bq + r, 0 = r < b dks larq"V djrs gSaA vkius ;g Hkh le> fy;k gksxk fd ;g
yach foHkktu izfØ;k osQ vfrfjDr oqQN Hkh ugha gS] ftls vki brus o"kks± rd djrs pys
vk, gSa rFkk q vkSj r dks Øe'k% HkkxiQy (quotient) vkSj 'ks"kiQy (remainder) dgk tkrk gSA .
bl ifj.kke dk vkSipkfjd dFku fuEufyf[kr gS%
izes; 1.1 (;wfDyM foHkktu izesf;dk) : nks /ukRed iw.kk±d a vkSj b fn, jgus ij] ,slh
vf}rh; iw.kZ la[;k,¡ q vkSj r fo|eku gSa fd a = bq + r, 0 = r < b gSA
bl ifj.kke dh tkudkjh laHkor% cgqr igys le; ls Fkh] ijarq fyf[kr :i esa bldk
loZizFke mYys[k ;wfDyM ,yhesaV~l (Euclid's Elements) dh iqLrd VII esa fd;k x;kA
;wfDyM foHkktu ,YxksfjFe (dyu fof/) blh izesf;dk (Lemma) ij vk/kfjr gSA
,YxksfjFe lqifjHkkf"kr pj.kksa dh ,d  Ük`a[kyk gksrh
gS] tks ,d fo'ks"k izdkj dh leL;k dks gy djus
dh ,d izfØ;k ;k fof/ iznku djrh gSA
'kCn ^,YxksfjFke* 9oha 'krkCnh osQ ,d iQkjlh
xf.krK vy&[okfjT
+
keh osQ uke ls fy;k x;k gSA
okLro esa] 'kCn ^,ytcjk* (Algebra) Hkh bUgha dh
fyf[kr iqLrd ^fglkc vy&T
+
kcj ok vy eqdkcyk*
ls fy;k x;k gSA
izesf;dk ,d fl¼ fd;k gqvk dFku gksrk gS
vkSj bls ,d vU; dFku dks fl¼ djus esa iz;ksx
djrs gSaA
;wfDyM foHkktu ,YxksfjFe nks /ukRed iw.kk±dksa dk HCF ifjdfyr djus dh ,d
rduhd gSA vkidks ;kn gksxk fd nks /ukRed iw.kk±dksa  a vkSj b dk HCF og lcls cM+k
iw.kk±d d  gS] tks  a vkSj b nksuksa dks (iw.kZr;k) foHkkftr djrk gSA
eqgEen bCu ewlk vy&[okfjT
+
keh
(780 – 850 lk-;q-)
2018-19
Page 5


okLrfod la[;k,¡ 1
1
1.1 Hkwfedk
d{kk 9 esa] vkius okLrfod la[;kvksa dh [kkst izkjaHk dh vkSj bl izfØ;k ls vkidks
vifjes; la[;kvksa dks tkuus dk volj feykA bl vè;k; esa] ge okLrfod la[;kvksa
osQ ckjs esa viuh ppkZ tkjh j[ksaxsA ;g ppkZ ge vuqPNsn 1-2 rFkk 1-3 esa /ukRed iw.kk±dksa
osQ nks vfr egRoiw.kZ xq.kksa ls izkjaHk djsaxsA ;s xq.k gSa% ;wfDyM foHkktu ,YxksfjFe (dyu
fof/) (Euclid’s division algorithm) vkSj vadxf.kr dh vk/kjHkwr izes; (Fundamental
Theorem of Arithmetic) A
tSlk fd uke ls fofnr gksrk gS] ;wfDyM foHkktu ,YxksfjFe iw.kk±dksa dh foHkkT;rk
ls fdlh :i esa lacaf/r gSA lk/kj.k Hkk"kk esa dgk tk,] rks ,YxksfjFe osQ vuqlkj] ,d
/ukRed iw.kk±d a dks fdlh vU; /ukRed iw.kk±d b ls bl izdkj foHkkftr fd;k tk
ldrk gS fd 'ks"kiQy  r  izkIr gks] tks b ls NksVk (de) gSA vki esa ls vf/drj yksx 'kk;n
bls lkekU; yach foHkktu izfØ;k (long division process) osQ :i esa tkurs gSaA ;|fi ;g
ifj.kke dgus vkSj le>us esa cgqr ljy gS] ijarq iw.kk±dksa dh foHkkT;rk osQ xq.kksa ls lacafèkr
blosQ vusd vuqiz;ksx gSaA ge buesa ls oqQN ij izdk'k Mkysaxs rFkk eq[;r% bldk iz;ksx
nks /ukRed iw.kk±dksa dk egÙke lekiorZd (HCF) ifjdfyr djus esa djsaxsA
nwljh vksj] vadxf.kr dh vk/kjHkwr izes; dk laca/ /ukRed iw.kk±dksa osQ xq.ku ls
gSA vki igys ls gh tkurs gSa fd izR;sd HkkT; la[;k (Composite number) dks ,d
vf}rh; :i ls vHkkT; la[;kvksa (prime numbers) osQ xq.kuiQy osQ :i esa O;Dr fd;k
tk ldrk gSA ;gh egRoiw.kZ rF; vadxf.kr dh vk/kjHkwr izes; gSA iqu%] ;g ifj.kke
dgus vkSj le>us esa cgqr ljy gS] ijarq blosQ xf.kr osQ {ks=k esa cgqr O;kid vkSj lkFkZd
vuqiz;ksx gSaA ;gk¡] ge vadxf.kr dh vk/kjHkwr izes; osQ nks eq[; vuqiz;ksx ns[ksaxsA ,d
okLrfod la[;k,¡
2018-19
2 xf.kr
rks ge bldk iz;ksx d{kk IX esa vè;;u dh xbZ oqQN la[;kvksa] tSls 2, 3 vkSj 
5
vkfn dh vifjes;rk fl¼ djus esa djsaxsA nwljs] ge bldk iz;ksx ;g [kkstus esa djsaxs fd
fdlh ifjes; la[;k] eku yhft, ( 0)
p
q
q
? , dk n'keyo izlkj dc lkar (terminating)
gksrk gS rFkk dc vlkar vkorhZ (non-terminating repeating) gksrk gSA ,slk ge 
p
q
osQ gj
q osQ vHkkT; xq.ku[kaMu dks ns[kdj Kkr djrs gSaA vki ns[ksaxs fd q osQ vHkkT; xq.ku[kaMu
ls 
p
q
osQ n'keyo izlkj dh izo`Qfr dk iw.kZr;k irk yx tk,xkA
vr%] vkb, viuh [kkst izkjaHk djsaA
1.2 ;wfDyM foHkktu izesf;dk
fuEufyf[kr yksd igsyh* ij fopkj dhft,%
,d foozsQrk lM+d ij pyrs gq, vaMs csp jgk FkkA ,d vkylh O;fDr] ftlosQ ikl
dksbZ dke ugha Fkk] us ml foozsQrk ls oko~Q&;q¼ izkjaHk dj fn;kA blls ckr vkxs c<+ xbZ
vkSj mlus vaMksa dh Vksdjh dks Nhu dj lM+d ij fxjk fn;kA vaMs VwV x,A fooszQrk us
iapk;r ls dgk fd ml O;fDr ls VwVs gq, vaMksa dk ewY; nsus dks dgsA iapk;r us fooszQrk
ls iwNk fd fdrus vaMs VwVs FksA mlus fuEufyf[kr mÙkj fn;k%
nks&nks fxuus ij ,d cpsxk_
rhu&rhu fxuus ij nks cpsaxs_
pkj&pkj fxuus ij rhu cpsaxs_
ik¡p&ik¡p fxuus ij pkj cpsaxs_
N%&N% fxuus ij ik¡p cpsaxs_
lkr&lkr fxuus ij oqQN ugha cpsxk_
esjh Vksdjh eas 150 ls vf/d vaMs ugha vk ldrsA
vr%] fdrus vaMs Fks\ vkb, bl igsyh dks gy djus dk iz;Ru djsaA eku yhft,
vaMksa dh la[;k a gSA rc mYVs Øe ls dk;Z djrs gq,] ge ns[krs gSa fd a la[;k 150 ls
NksVh gS ;k mlosQ cjkcj gSA
;fn lkr&lkr fxusa] rks oqQN ugha cpsxkA ;g a = 7p + 0 osQ :i esa ifjo£rr gks tkrk
gS] tgk¡ p dksbZ izko`Qr la[;k gSA
* ;g ^U;wesjslh dkmaV~l* (ys[kdx.k ,- jkeiky vkSj vU;) esa nh igsyh dk ,d ifjo£rr :i gSA
2018-19
okLrfod la[;k,¡ 3
;fn N%&N% fxusa] rks 5 cpsaxsA ;g a = 6q + 5 osQ :i esa ifjo£rr gks tkrk gS] tgk¡
q dksbZ izko`Qr la[;k gSA
ik¡p&ik¡p fxuus ij] 4 cpsaxsA ;g a = 5s + 4 esa ifjo£rr gks tkrk gS] tgk¡ s dksbZ
izko`Qr la[;k gSA
pkj&pkj fxuus ij] 3 cpsaxsA ;g a = 4t + 3, esa ifjo£rr gks tkrk gS] tgk¡ t dksbZ
izko`Qr la[;k gSA
rhu&rhu fxuus ij 2 cpsaxsA ;g a = 3u + 2 esa ifjo£rr gks tkrk gS] tgk¡ u dksbZ
izko`Qr la[;k gSA
nks&nks fxuus ij] 1 cpsxkA ;g a = 2v + 1, esa ifjo£rr gks tkrk gS tgk¡ v dksbZ izko`Qr la[;k gSA
vFkkZr~] mijksDr izR;sd fLFkfr esa] gekjs ikl nks /ukRed iw.kk±d a vkSj b gSa (fy, x,
mnkgj.k esa b osQ eku Øe'k% 7, 6, 5, 4, 3 vkSj 2 gSa)A buesa a dks b ls Hkkx nsus ij 'ks"k r
cprk gS (mijksDr esa r osQ eku Øe'k% 0, 5, 4, 3, 2 vkSj 1 gSa) vFkkZr~] r Hkktd b ls NksVk
gSA tSls gh ge bl izdkj osQ lehdj.k fy[krs gSa] ge ;wfDyM foHkktu izesf;dk
(Euclid’s division lemma) dk iz;ksx dj jgs gSa] ftls izes; 1-1 esa fn;k tk jgk gSA
vc viuh igsyh ij okil vkus ij] D;k vki dksbZ ckr lksp dj crk ldrs gSa
fd bl igsyh dks oSQls gy djsaxs\ gk¡! vki 7 osQ ,sls xq.ktksa dks [kksft, tks mijksDr lHkh
izfrca/ksa dks larq"V djasA tk¡p vkSj Hkwy fof/ ls (LCM dk iz;ksx djosQ) vki Kkr dj
ldrs gSa fd vaMksa dh la[;k 119 FkhA
bl ckr dk vuqHko djus osQ fy, fd ;wfDyM foHkktu izesf;dk D;k gS] iw.kk±dksa
osQ fuEufyf[kr ;qXeksa ij fopkj dhft,%
(i) 17, 6 (ii) 5, 12 (iii) 20, 4
tSlk fd geus igsyh okys mnkgj.k esa fd;k Fkk] ;gk¡ Hkh ge izR;sd ;qXe osQ fy,
laca/ fy[k ldrs gSa tSlk fd uhps n'kkZ;k x;k gSA
(i) 17 = 6 × 2 + 5 (17 esa 6 nks ckj tkrk gS vkSj 'ks"k 5 cprk gS)
(ii) 5 = 12 × 0 + 5 (;g laca/ blfy, lgh gS] D;ksafd 12] 5 ls cM+k gS)
(iii) 20 = 4 × 5 + 0 (20 esa 4 ik¡p ckj tkrk gS vkSj oqQN 'ks"k ugha cprk)
vFkkZr~ /ukRed iw.kk±dksa  a vkSj b osQ izR;sd ;qXe osQ fy,] geus ,slh iw.kZ la[;k,¡  q
vkSj r Kkr dj pqosQ gSa fd
a = bq + r, 0 = r < b gSA
2018-19
4 xf.kr
è;ku nhft, fd  q  ;k  r 'kwU; Hkh gks ldrs gSaA
vc vki /ukRed iw.kk±dksa  a vkSj b osQ fuEufyf[kr ;qXeksa osQ fy, iw.kk±d q vkSj r
Kkr djus dk iz;Ru dhft,%
(i) 10, 3 (ii) 4, 19 (iii) 81, 3
D;k vki è;ku ns jgs gSa fd q vkSj r vf}rh; gSa\ ;s gh osQoy ,sls iw.kk±d gSa] tks
izfrca/ksa  a = bq + r, 0 = r < b dks larq"V djrs gSaA vkius ;g Hkh le> fy;k gksxk fd ;g
yach foHkktu izfØ;k osQ vfrfjDr oqQN Hkh ugha gS] ftls vki brus o"kks± rd djrs pys
vk, gSa rFkk q vkSj r dks Øe'k% HkkxiQy (quotient) vkSj 'ks"kiQy (remainder) dgk tkrk gSA .
bl ifj.kke dk vkSipkfjd dFku fuEufyf[kr gS%
izes; 1.1 (;wfDyM foHkktu izesf;dk) : nks /ukRed iw.kk±d a vkSj b fn, jgus ij] ,slh
vf}rh; iw.kZ la[;k,¡ q vkSj r fo|eku gSa fd a = bq + r, 0 = r < b gSA
bl ifj.kke dh tkudkjh laHkor% cgqr igys le; ls Fkh] ijarq fyf[kr :i esa bldk
loZizFke mYys[k ;wfDyM ,yhesaV~l (Euclid's Elements) dh iqLrd VII esa fd;k x;kA
;wfDyM foHkktu ,YxksfjFe (dyu fof/) blh izesf;dk (Lemma) ij vk/kfjr gSA
,YxksfjFe lqifjHkkf"kr pj.kksa dh ,d  Ük`a[kyk gksrh
gS] tks ,d fo'ks"k izdkj dh leL;k dks gy djus
dh ,d izfØ;k ;k fof/ iznku djrh gSA
'kCn ^,YxksfjFke* 9oha 'krkCnh osQ ,d iQkjlh
xf.krK vy&[okfjT
+
keh osQ uke ls fy;k x;k gSA
okLro esa] 'kCn ^,ytcjk* (Algebra) Hkh bUgha dh
fyf[kr iqLrd ^fglkc vy&T
+
kcj ok vy eqdkcyk*
ls fy;k x;k gSA
izesf;dk ,d fl¼ fd;k gqvk dFku gksrk gS
vkSj bls ,d vU; dFku dks fl¼ djus esa iz;ksx
djrs gSaA
;wfDyM foHkktu ,YxksfjFe nks /ukRed iw.kk±dksa dk HCF ifjdfyr djus dh ,d
rduhd gSA vkidks ;kn gksxk fd nks /ukRed iw.kk±dksa  a vkSj b dk HCF og lcls cM+k
iw.kk±d d  gS] tks  a vkSj b nksuksa dks (iw.kZr;k) foHkkftr djrk gSA
eqgEen bCu ewlk vy&[okfjT
+
keh
(780 – 850 lk-;q-)
2018-19
okLrfod la[;k,¡ 5
vkb, lcls igys ,d mnkgj.k ysdj ns[ksa fd ;g ,YxksfjFe fdl izdkj dk;Z
djrk gSA eku yhft, gesa iw.kk±dksa 455 vkSj 42 dk HCF Kkr djuk gSA ge cM+s iw.kk±d
455 ls izkjaHk djrs gSaA rc ;wfDyM izesf;dk ls] gesa izkIr gksrk gS%
455 = 42 × 10 + 35
vc Hkktd 42 vkSj 'ks"kiQy 35 ysdj] ;wfDyM izesf;dk dk iz;ksx djus ij] gesa
izkIr gksrk gS%
42 = 35 × 1 + 7
vc] Hkktd 35 vkSj 'ks"kiQy 7 ysdj] ;wfDyM izesf;dk dk iz;ksx djus ij] gesa
izkIr gksrk gS%
35 = 7 × 5 + 0
è;ku nhft, fd ;gk¡ 'ks"kiQy 'kwU; vk x;k gS rFkk ge vkxs oqQN ugha dj ldrsA
ge dgrs gSa fd bl fLFkfr okyk Hkktd] vFkkZr~ 7 gh 455 vkSj 42 dk HCF gSA vki
bldh lR;rk dh tk¡p 455 vkSj 42 osQ lHkh xq.ku[kaMksa dks fy[kdj dj ldrs gSaA ;g
fof/ fdl dkj.k dk;Z dj tkrh gS\
bldk dkj.k ;wfDyM foHkktu ,YxksfjFe gS] ftlosQ pj.kksa dks uhps Li"V fd;k
tk jgk gS%
nks /ukRed iw.kk±dksa] eku yhft, c vkSj d (c > d) dk HCF Kkr djus osQ fy, uhps
fn, gq, pj.kksa dk vuqlj.k dhft,%
pj.k 1 : c vkSj  d osQ fy, ;wfDyM foHkktu izesf;dk dk iz;ksx dhft,A blfy,] ge ,sls
q vkSj  r  Kkr djrs gSa fd c = dq + r, 0 = r < d gksA
pj.k 2 : ;fn  r = 0 gS] rks  d  iw.kk±dksa  c vkSj d dk HCF gSA ;fn r ? 0 gS] rks
d vkSj  r  osQ fy,] ;wfDyM foHkktu izesf;dk dk iz;ksx dhft,A
pj.k 3 : bl izfØ;k dks rc rd tkjh jf[k,] tc rd 'ks"kiQy 0 u izkIr gks tk,A blh
fLFkfr esa] izkIr Hkktd gh okafNr HCF gSA
;g ,YxksfjFe blfy, izHkko'kkyh gS] D;ksafd HCF  (c, d) = HCF (d, r) gksrk gS] tgk¡
laosQr HCF (c, d) dk vFkZ gS c vkSj d dk HCFA
mnkgj.k 1 : 4052 vkSj 12576 dk HCF ;wfDyM foHkktu ,YxksfjFe dk iz;ksx djosQ
Kkr dhft,A
2018-19
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