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Texte intégral

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(3) lk]t[ko |hk. ສດິ. d 0 ! !W !( @& ## #) $* &* *& (# )! )( _$. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 7eoe lk]t[ko [qfmu ! 9eo;ome,t-kf [qfmu @ 9eo;o4h;o [qfmu # dko[;d c]t dko]q[9eo;ome,t-kf [qfmu $ dko[;d c]t dko]q[9eo;o4h;o [qfmu & dko76o c]t dkosko9eo;ome,t-kf [qfmu * dko76o c]t dkosko9eo;o4h;o [qfmu ( mt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f 0v’9eo;ome,t-kf [qfmu ) 9eo;orNo4ko8jk’M [qfmu _ 7jkle[6o [qfmu !W leo;og]d7toyf [qfmu !! 7;k,leobddjP;da[g,af c]t glAo-nj [qfmu !@ glAo-nj0t|ko c]t glAo-nj8A’lkd [qfmu !# dkoc8h,glAo-nj0t|ko c]t glAo-nj8A’lkd [qfmu !$ dko[;d];’pk;mjvo-nj c]t dko76o9eo;.f|bj’ da[];’pk;0v’mjvo-nj [qfmu !& dko;afcmd];’pk;0v’mjvo-nj [qfmu !* dko;afcmd];’pk; [qfmu !( dkog7yj’7n [qfmu !) ,5,F dko;afcmd,5,. 0. !W! !W) !!* !@_ !#*.

(4) 7. !$( !&* !*( !(& !)# !)) !_# !_) @W* @!! @@# @## @#_. ສດິ. I6[g7yj’7n0v’,5, c]t 0v’glAo-nj cdog7yj’7n0v’I6[F glAodk’lkdF glAoc[j’g7yj’,5, ,kf8klj;o ,kf8klj;o X8+? dj5, c]t vq’xtdv[0v’d5j, g7jnv’\kp ∈ c]t ∉ c]t d5j,gmqjkdao vto5d5j, c]t czo;kf0v’d5j, dko8afdao0v’d5j,{dko3I,dao0v’d5j, vaf8klj;o vaf8klj;orq;rao -vdskrqfmuj[+maoI6h0v’vaf8klj;orq;rao dko7yfw]jlj;oIhvp vaf8klj;orq;raoxUo. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. [qfmu !_ [qfmu @W [qfmu @! [qfmu @@ [qfmu @# [qfmu @$ [qfmu @& [qfmu @* [qfmu @( [qfmu @) [qfmu @_ [qfmu #W [qfmu #!.

(5) rkdmu ! 9eo;ome,t-kf gg]t 9eo;o4h;o [qfmu ! 9eo;ome,t-kf dyf9tde dyf9tdemu ! dD 9qj’8nj,9eo;ooadIPo.oshv’0v’g9Qk.lj[jvo9Eg,afD 16j3I’IPo,afmtpq,csj’|bj’gryjowfhltcf’lt4y8y9eo;ooadIPoc8j]tshv’ faj’8k8t]k’]5j,oUD ,!D 0 $W @@. ,!D 7 #) @!. ,!D ’ $! @#. shv’g9Qkgv’ DDDD DDDD. ສດິ. shv’ ,!D d 9eo;oma’\qf #& py’ !_. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !D. 0D 9qj’oa[9eo;o\kd[u16jc8j]tdt8jkc]h;8nj,.lj[jvo9Eg,afD. ,u DDDDDDD |j;p. ,u DDDDDDD |j;p. ,u DDDDDDD |j;p. ,u DDDDDDD |j;p. 7D 9eo;omuj.-h[aombd s^n .-h.odkooa[ g-aoG 9eo;ooadIPoc8j]tshv’ c]t 9eo;o \kd[u16jc8j]tdt8jk gryjogvUo;jkG 9eo;ome,t-kfD 9qj’pqdgsfdkomuj9egxao8hv’.-h9eo;ome,t-kfgrnjv[aombd,k9adlv’8q;1jk’D ’D gryjo[aombdzqodko;afcmdoE|ad c]t ];’l6’0v’oadIPo 3fpmjq;wx9eo;o muj.-hc,jo9eo;oco;.fL. 1.

(6) dyf9tdemu @ 9eo;o.f|bj’ lk,kf0Pogxao8q;g]dS 8q;|a’ln X7evjko?D dD dko0Po9eo;o.oI6[c[[dt9kpD 8q;1jk’G 41 375 = 41×1 000 + 3×100 + 7×10 + 5 vjko;jkG ljuly[gvafraolk,Ihvpg9afly[shkD 9qj’vu’.lj8q;1jk’c]h;0Po9eo;o]5j,oU.oI6[c[[dt9kp c]h;vjkorhv,D 507; 2532; 137 065; 45 700; 3 512 415; 25 718 272; 3 125 637 542.. 0D dko0Po9eo;ogxao8q;g]dD 8q;1jk’G lujraogdQkIhvpshkly[g9af = 4 ×1000 + 9 ×100 + 5 ×10 + 7 = 4957. dyf9tdemu # dkoxP[mP[9eo;o3fpoe.=-hg7njv’\kp <F >F = 8q;1jk’G. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 9qj’vu’.lj8q;1jk’c]h;0Po9eo;o]5j,oUgxao8q;g]dD { -k;gdQkraocxfly[ { lv’Ihvplujly[g9afraosqdIhvplk,ly[gvaf { shk]hkog9afly[lk,rao|bj’Ihvpcxfly[sqd { ly[lv’8NljuIhvp-k;lv’]hkoshkIhvpgdQkly[g9afraocxfIhvpsqdly[sqd { @!# 8N (& ]hko #() rao &$WD. 4 × 1000 + 9 × 100 + 5 × 10 + 6 = 4956 2 × 1000 + 4 × 100 + 3 × 10 + 5 < 2 × 1000 + 4 × 100 + 3 × 10 + 7. 9qj’vu’.lj8q;1jk’c]h;xP[mP[9eo;o]5j,oU 3fp0Pog7njv’\kp <F >F = .lj[jvo 9Eg,afD dD 2 ×1000 + 4 ×100 + 3 ×10 + 1 ....... 2430 0D 2 ×1000 + 5 ×100 + 7 ×10 + 5 ....... 2575 7D 4 ×1000 + 9 ×100 + 5 ×10 + 6 ....... 5 ×1000 + 9 ×100 + 5 ×10 + 6 ’D 6 ×1000 + 2 ×100 + 5 ×10 + 8 ....... 6 ×1000 + 2 ×100 + 4 ×10 + 8 2.

(7) dyf9tdemu $ dD 9qj’0ufmjvo-nj ]5j,oU3fpg]uj,9kd95fmujdeoqf.shfaj’I6[ wx8k,mk’ovog[Nv’0;kD [AB] pk; # cm A [MN] pk; & cm M [EF] pk; & cm E [PQ] pk; ) cm P [RS] pk; !W cm. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 0D 9kdI6[c8h,F 9qj’xP[mP[];’pk;0v’mjvo-njgs^qjkoAoD 8q;1jk’G mjvo-nj [MN] pk;d;jkmjvo-nj [AB] faj’oAo gryjog;Qk;jkG & s^kpd;jk #S 0PoG & > #S xUo7noG mjvo-nj [AB] lAod;jk [MN]. faj’oAo gryjog;Qk;jkG # |hvpd;jk & 0PoG # < &D 9qj’xP[mP[];’pk;0v’c8j]tlv’mjvo-nj8+wxoUG [MN] c]t [EF]S [EF] c]t [PQ]S [PQ] c]t [RS]; [PQ] c]t [MN]S [AB] c]t [EF]S [AB] c]t [RS]. ສດິ. R. @D. .97;k,D !D 9eo;ome,t-kfD { 9eo;og-yj’gryjo.-hoa[ s^n [aombd9eo;ooadIPoc8j]tshv’ c]t 9eo;o\kd[u16j c8j]tdt8jkg-ajoG #&S !_S $WS @@S #)S @!S $!S @#S @@S !)S @WS @$ gvUo;jkG 9eo;ome,t-kf { 9eo;ome,t-kfma’\qf3I,g0Qkdao gxaod5j,|bj’,u-nj;jkdj5,9eo;ome,t-kf c]t laopt]adfh;p8q;vadlvo ` D gryjo0Po ` = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10;…} { gryjo.-hg7njv’\kp ∈ cmo7e;jk <gxaovq’xtdv[0v’ < c]t ∉ cmo7e;jk < [+gxaovq’xtdv[0v’ < 3.

(8) ສ. sq;Ihສvp ວ. ະຫ ສ sq;ly[ ງວ ນລ ຂິ ະ sq;|j;p. sq;rao. sq;\jno. sq;clo. sq;]hko. sq;3df. sq;8N. ສດິ. 8q;1jk’G ! c,jo9eo;ome,t-kfF gryjog;Qk;jk ! gxaovq’xtdv[0v’d5j, ` . 0Posp=h ! ∈ ` #F& [+c,jo9eo;ome,t-kfF gryjog;Qk;jk #F& [+c,jovq’xtdv[0v’d5j, ` . 0Posp=h #F& ∉ ` @D dkovjko c]t dko0Po9eo;ome,t-kf 8k8t]k’dko0Po c]t dkovjko9eo;ome,t-kf -Ao0v’sq;8N -Ao0v’sq;]hko -Ao0v’sq;rao -Ao0v’sq;|j;p sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; Ihvp ly[ |j;p Ihvp ly[ |j;p Ihvp ly[ |j;p Ihvp ly[ |j;p ( @ & ( * & $ _ * $ # ) @ & _ * ) W W & # W ( @ W W ) W $. 572 = 5×100 + 7×10 + 2. vjko;jkG shkIhvpg9afly[lv’ 4 765 = 4×1000 + 7×100 + 6 ×10 + 5. vjko;jkG lujraog9afIhvpsqdly[shk 43 896 = 43 ×1000 + 8 ×100 + 9 ×10 + 6. vjko;jkG lujly[lk,raocxfIhvpgdQkly[sqd s^n luj\njolk,raocxfIhvpgdQkly[sqdD 2 596 800 = 2 +1000 000 + 596 ×1000 + 8 ×100. vjko;jkG lv’]hkoshkIhvpgdQkly[sqdraocxfIhvp 3 072 008 045 = 3×1000 000 000 + 72 × 1000 000 + 8 × 1000 + 4 × 10 + 5. vjko;jkG lk,8Ng9afly[lv’]hkocxfraolujly[shk 4.

(9) #D dkoxP[mP[9eo;ome,t-kf { dko9af]Po9eo;ome,t-kfmuj0Pofh;p|bj’8q;g]dc8j|hvpsks^kpfaj’oUG 1<2<3<4<5<6<7<8<9. { dkoxP[mP[9eo;ome,t-kfmuj,u9eo;o8q;g]d8jk’dao D 9eo;o.f,u9eo;o8q;g]ds^kpd;jkc,jo9eo;ooAos^kpd;jkD 8q;1jk’G !W c]t _ S !W ,ulv’8q;g]d c]t _ ,u|bj’8q;g]d ltoAo !W > 9 !W WWW c]t _ ___S _ ___ ,ulj8u q;g]d c]t !W WWW ,ushk8q;g]d ltoAo _ ___ < !W WWWD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. { dkoxP[mP[9eo;ome,t-kfmuj,u9eo;o8q;g]dgmqjkdaoD 3fpdkola’gdfc8j-hkpsk0;kF 9eo;o.f,u8q;g]dmuj|bj’s^kpd;jkc,jo 9eo;ooAos^kpd;jkD 8q;1jk’G ()$& c]t &_)(F lv’9eo;ooU0Pofh;pluj8q;g]d c]t 8q;g]dmuj|bj’ c,jo ( c]t & g-yj’;jk ( > &F ltoAo ()$& > 5987D d=]tou8q;g]dmuj|bj’gmqjkdaoF gryjo9txP[mP[8q;g]dmulv’ s^n mulk, c]t 8q;g]d 8+MwxD 8q;1jk’G *)!# c]t *&_)F gIqkgsao;jk8q;g]dmuj|bj’gmqjkdaoD faj’oAogIqkmP[8q;g]dmujlv’D gIqkgsao;jk ) > & ltoAo * )!# > * &_)D @ #)( *&W c]t @ #)_ !(&F gIqkgsao;jk8q;g]dmu|bj’F mulv’F mulk,gmqjkdaoF gIqkmP[8q;g]dmuluj gsao;jk 7 < 9 ltoAo @ #)( *&W < @ #)_ !(&D. { dko0v[0Ao9eo;ome,t-kf gryjo0v[0Ao9eo;o.f|bj’fh;p7jk.dh7P’s^5f c]t 7jk.dh7P’]njo16jsâd8jk’M 0v’,aoD 8q;1jk’G gryjo0v[0Ao _ ()# 16jsâd8jk’M faj’oUG 7jk.dh7P’s^5f 7jk.dh7P’]njo 16jsq;|j;p _ ()@ < _ ()# < _ ()$ 16jsq;ly[ _ ()W < _ ()# < _ (_W 16jsq;Ihvp _ (WW < _ ()# < _ )WW. 5.

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(11) #D. 9qj’0Po9eo;o8+wxoUgxao8q;|a’ln.olv’I6[c[[8jk’daoD 8q;1jk’G @ W#& )WW (&!D { lv’8Nlk,ly[shk]hkocxfIhvpraog9afIhvpshkly[gvafD { lv’8Nlk,3dfshk]hkocxfclog9afIhvpshkly[gvafD dD !& @)# $((S ’D !W &(* #_WD. 0D # !_W )(& $@WS 9D !@ _(# !&( *@_D. 7D @( &)@ W&!S. 9qj’.-hshk8q;g]d8+wxoUG #F !F WF WF @ 0Po9eo;omuj,ushk8q;g]d grnjv.shwfh9eo;omuj,u 7jks^kpl5f c]t 9eo;omuj,u7jk|hvpl5fD. &D. 9qj’xtdv[8k8t]k’]5j,oU.sh4nd8hv’g\ktlq,D 8k,4aoG dD 8q;g]dsq;|j;pc,jo # c]t 8q;g]dsq;Ihvpc,jo @D 0D 8q;g]dsq;|j;pc,jo ) c]t gxao9eo;o|hvpd;jk !WW 7D 8q;g]dsq;|j;pc,jo WD ’D 8q;g]dsq;raoc,jo @ c]t 8q;g]dsq;Ihvpgmqjklv’gmjnv0v’8q;g]dsq;rao. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. $D. 8k,c4;G A. 8q;g]dsq;raoc,jo # c]t 8q;g]dsq;ly[c,jo (D BD 8q;g]dsq;Ihvpc,jo W c]t 8q;g]dsq;ly[c,jo ) CD $ V !WW + & DD 8q;g]dsq;rao c]t 8q;g]dsq;|j;pc,jo8q;g]dfP;daoD d. 0. 7. ’. A B C D. 7.

(12) 9qj’-vd|bj’8q;g]d8k,.9 ,kxjPocmo8q;g]d.f|bj’.o9eo;o @( $)@ grnjv.shwfhG dD 9eo;o,ushk8q;g]d,u7jks^kpl5fD 0D 9eo;o,ushk8q;g]d,u7jk|hvpl5fD. (D 9qj’xP[mP[9eo;oc8j]t76j]5j,oUfh;pg7njv’\kp <; > s^n = D dD $ #)( c]t # __)S 0D &( @$) c]t &( $)@S 7D @ W!& WW! c]t !W WW# WWWS ’D ! WWW WW! c]t ___ ___ 9D (& W&# **( c]t (& W&@ **(S lD ! (WW c]t _ V !WW + ( V !W + & -D !& ]hko #$& rao (#@ c]t |bj’3dfg9af]hkogdQkIhvpraogvafD pD # V !W WWW + & V !WW + !W + ) c]t #W &!@D fD @ V !WWW + ( V !W + _ c]t @ V !W WWW + ( V !W + &D. ສດິ. *D. 9qj’la[xjPo[jvo0v’8q;g]dmuj.-h0Po9eo;o )!* _@# (_) rP’7A’fP;grnjv.shwfhG dD 9eo;owfh,kgxao9eo;o,u7jks^kpl5fmuj0Pofh;p8q;g]dgs^qjkoAoD 0D 9eo;owfh,kgxao9eo;o|hvpl5fmuj0Pofh;p8q;g]dgs^qjkoAoD. _D. 9qj’djkp7no9eo;o8+wxoU3fpgvqk8q;g]d W muj16jsâd[+,u7;k,\kpvvdF c]h;9af]Po 9eo;ofaj’djk;c8j|hvpsks^kpD dD _WWS WW_S _W!S !_WS W_!S WW!_S !WW_S _W!WS _!WWS _WW!D 0D WW#W$!S WW#$!WS W#!WW$S WW$!#WS W$#!WWS WWW$#!S $WWW#!D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. )D. !WD 9qj’9af]Po9eo;o]5j,oUc8js^kpsk|hvpD dD ( *$@S * (@$S $ (*@S * ($@S @ $*(S $*@(S ( $*@S $ @*(S @ ($*S $ @(*D 0D @* W#&S *# &W@S @* &#WS @# *&WS @# &W*S @# &*WS &W *#@S &* W#(D. 8.

(13) !!D. 9qj’0v[0Ao9eo;o]5j,oU fh;p7jk.dh7P’s^5f c]t 7jk.dh7P’]njo16jsâd8jk’MD dD DDDDDDDDDDDDDDDDD < ()$# < DDDDDDDDDDDDDDDDD 16jsq;|j;pD 16jsq;IhvpD 0D DDDDDDDDDDDDDDDDD < !()&W < DDDDDDDDDDDDDDDDD 7D DDDDDDDDDDDDDDDDD < _)!(# < DDDDDDDDDDDDDDDDD 16jsq;ly[D ’D DDDDDDDDDDDDDDDDD < ! @)( *WW < DDDDDDDDDDDDDDDDD 16jsq;raoD 16jsq;\njoD 9D DDDDDDDDDDDDDDDDD < ! @)( *WW < DDDDDDDDDDDDDDDDD !@D w,h]e|bj’pk; &_#( mm. 9qj’0v[0Ao];’pk;0v’w,h]eoUfh;psq;|j;p m c]t cmD g]d39fG !#D I6h;jk ! xu ,u #*& ;aoS ! ;ao ,u @$ -qj;3,’S ! -qj;3,’ ,u # *WW ;yokmuD 4k,;jk 7qomuj,u vkp5 (& xu ,uvkp5pnog4y’8N;yokmu s^n [+ L. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !&D .shI6h ! 38ogmqjk !WWW kg c]t ! kg gmqjk !WWW gD [=]ylafzt]yf1kltzq,csj’. ສດິ. !$D w]ptsjk’c8j|hk3]dsk8kg;ao,u !&W ]hkody3]c,af c]t 7;k,w;0v’cl’ c,jo #WW WWW km /s. 4k,;jk cl’8kg;ao.-hg;]k9ad;yokmugrnjv,kg4y’|hk3]d L |bj’.-hw0jgs^nv’ #WW g 8+1kltzq, ! 38oD 9qj’7yfw]j,;olko0v’w0jgs^nv’.o1k ltzq, @W kgD. !*D ;af45vao|bj’|ad (& g S ;af45-toyffP;daooU]hkovao9t|adgmqjk.f mg L. !(D w,h0ufw2vao|bj’pk; & cmD 4hkgryjogvqkw,h0ufw2-toyffP;daooU8+dao3dfvao9tpk; 9addy3]c,af L !)D g]d3ic,’D dD 9qj’0Po9eo;o]5j,oUfh;pg]d3ic,’D *_S (!S _(S #$(S *&(S )_$S # (*$S ! &@_S *W(S ! !W)D 0D 9qj’0Po9eo;o]5j,oUfh;pg]dvkia[D CXLIX; CDLXXVIII; MCDLXIV; MMCCCXLIV; MMMCMXCIX; ICVIII;. CVI;. MI;. MXIII;. 9. MVII..

(14) [qfmu @ 9eo;o4h;o dyf9tde dyf9tdemu ! mhk; 7e g9Qk0v’Ihko0kpvkskol5dcsj’|bj’ wfh[aombd]kpIa[ { ]kp9jkpxt9e ;ao0v’]k;.o|bj’vkmyffaj’oUD 0kpwfhma’\qf ]kp9jkpma’\qf zqolts^5[xt9e;ao Xdu[? Xdu[? Xdu[? ;ao9ao )(& W&W *#W W(W + @$$ _)W ;aova’7ko &W_ &)& &)( W&W – (( $*& ;aor5f &@# &&W &@! W#W ;aortsaf *$& &(W *&@ (WW ;aol5d (#W &(& &$# (@& ;aoglqk ! W#& ()W )#* (_W ;aovkmyf ! WW( )&W )W_ &$W. ສດິ. ,N. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !. wfh7yfw]jzqolts^5[xt9e;ao faj’oUG ;ao9aoG 0kpwfhs^kpd;jk]kp9jkpF ]k;wfhdew]D zqolts^5[\kp[;d 16j|hkzqo]q[ ]ts;jk’9eo;og’yo0kpwfhma’\qf .sh9eo;og’yo]kp9jkpma’\qfD ;aova’7koG 0kpwfh|hvpd;jk]kp9jkpF ]k;0kfmboF zqolts^5[\kp]q[ 16j|hkzqo]q[ ]ts;jk’]kp9jkpma’\qf .sh9eo;og’yo0kpwfhma’\qfD dD 9qj’djkp7no8k8t]k’gmy’oU c]h;8nj,.lj.sh4nd8hv’D 0D vu’.lj8k8t]k’mujwfh9kd0=h Xd? c]h;[vd;jk ,u;ao.fcfjmujmhk; 7e wfhdew]D 7D le]a[dko7hk0kpF wfhdew] s^n 0kfmbo c,jolv’gsfdkodq’dao0hk,dao.o lt4kotdkofP;daoD grnjv[aombdlv’gsfdkofh;p9eo;o gryjodeoqfgvqkg7njv’ \kp[;d <+< -uh[vd;jkwfhdew] c]t g7njv’\kp]q[ <–< -uh[vd;jk0kfmboD. 10.

(15) 9eo;omuj,ug7njv’\kp <+< 16j8+|hkc,jo9eo;o[;d c]t 9eo;omuj,ug7njv’\kp <–< 16j8+|hk c,jo9eo;o]q[D 8q;1jk’G + !W vjko;jk [;dly[ c]t – !@ vjko;jk ]q[ly[lv’D 9qj’[vd9eo;o8=jwxoUF 9eo;o.fgxao9eo;o[;d c]t 9eo;o.fgxao9eo;o]q[D + &S + $S –13; + *; +!(S + 84; – 135; – 7; + 92; – 191; + 207; + 539;. – 26;. – 413D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. O. ສດິ. dyf9tdemu @ ok’ f;’fu pjk’8k,glAo-njfh;p[kfpjk’lt\jeltg\u3fpg]uj,8Qo9kdg,af o. dD 4hkgryjo\kpIvp8uo0v’]k;.ljglAo-njoAo fh;p8q;vadlvofaj’oUG pjk’wx|hk \kpfh;p A; B; C;……. ; 10 [kfdhk;D pjk’4vpsâ’ \kpfh;p A'; B'; C';……. ; 10 [kfdhk;D 9qj’ln[8+\kpIvp8uo0v’ ok’ f;’fu 16jglAo-njoAo.sh7q[4h;o.oc8j]tmyfmk’D. 0D grnjv-U[vdmyfmk’.odkopjk’ gryjo.-h9eo;o\kp]efa[Ivp8uo faj’oUG \kp W 16j95fg]uj,8QoF + ! 16j95fIvp8uopjk’wx|hk dhk;mu|bj’F + @ 16j95f[kf dhk;mujlv’ c]t 8+M wxD \kp –! 16j95fIvp8uopjk’4vpsâ’dhk;mu|bj’F –@ 16j95f Ivp8uopjk’4vpsâ’dhk;mulv’ c]t 8+MwxD 9qj’ln[8+deoqfIvp8uo0v’ ok’ f;’fu fh;p9eo;oD 7D gIqkla’gdfgsao;jkg,af A dq’da[ + !D faj’oAogryjo;jkg,af A 4nddeoqffh;p + ! gryjogvUo + ! ;jk< va[-yf < 0v’g,af A 0Posp=h A (+1). 9qj’[vdva[-yf0v’g,af8jk’M mujgIqk\kp16jglAo-njoAoD. #. .97;k, !D 9eo;o4h;o 9eo;ome,t-kfmuj,ug7njv’\kp < + < 16j8+|hkgvUo;jk 9eo;o4h;o[;dF sn^ 4hk[+,u g7njv’\kpd=gvUo;jk 9eo;o4h;o[;dF faj’oAo 9eo;o4h;o[;dc,jo 1; 2; 3; 4; DDD 11.

(16) @D cdo9eo;o4h;o. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. d5j,9eo;o4h;o[;dlaopt]adfh;p ]+ D ]+ = { 1; 2; 3; 4; DDD } 9eo;ome,t-kfmuj,ug7njv’\kp < – < 16j8+|hkgvUo;jk 9eo;o4h;o]q[F faj’oAo 9eo;o4h;o]q[c,jo –1; –2; –3; –4; DDD d5j,9eo;o4h;o]q[laopt]adfh;p ]− D ]− = { –1; –2; –3; –4; DDD } 9eo;o W gxao9eo;o4h;o c8j[+gxao9eo;o4h;o[;d c]t [+gxao9eo;o4h;o]q[D 9eo;o4h;omuj[+gxao9eo;o4h;o]q[ c,jo 0; 1; 2; 3; 4; DDD 9eo;o4h;omuj[+gxao9eo;o4h;o[;d c,jo 0; –1; –2; –3; –4; DDD 9eo;o4h;oma’\qf3I,g0Qkdaogxaod5j,|bj’,u-nj;jkdj5,9eo;o4h;o c]t laopt]ad fh;p ] D ] = {DDD – 4; –3; –2; –1; 0; 1; 2; 3; 4; DDD } m5dM9eo;ome,t-kf c,jo9eo;o4h;o c]t laopt]adfh;p ` ⊂ ] D. glAo-njmujc8j]tg,af,uva[-yfgxao9eo;o4h;o gvUo;jk cdo9eo;o4h;o g-ajo glAo-nj]5j,oUG E C. O I. A B D. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10. { g,afg7Qk O ,uva[-yf W 0Po OXW? { g,af I ,uva[-yf ! 0Po ; I (1) g,af A (+3) c]t C (– 3) 16jc8j]trkd0v’g,afg7Qk c]t ,uw]ptsjk’ltg\umP[ .ljg,afg7Qk O faj’oAo9eo;o +3 c]t 9eo;o – 3 c,jolv’9eo;odq’dao0hk,daoD .sh a ∈ ] F −a c,jo9eo;odq’dao0hk,0v’ a D 8q;1jk’G − ( −3) = 3, − ( +3) = −3 lv’9eo;o4h;odq’dao0hk,daoc,jo9eo;ofP;daoc8j,ug7njv’\kp8jk’daoD. 12.

(17) #D dkoxP[mP[9eo;o4h;o I6h;jk 9eo;o4h;o[;d c]t 9eo;ome,t-kfc,jo9eo;ovaofP;daoF dkoxP[ mP[9eo;o4h;o[;dd=7nda[dkoxP[mP[9eo;ome,t-kfD 9eo;o[;ds^kpd;jk9eo;o]q[g]NvpMD le]a[lv’9eo;o4h;o]q[F 9eo;o4h;o]q[.fmuj,uw]ptsjk’9kdg,afg7Qks^kpd;jk 9t,u7jk|hvpd;jkD 8q;1jk’G – & ,uw]ptsjk’9kdg,afg7Qk & sq;|j;p c]t – # ,uw]ptsjk’9kdg,af g7Qk # sq;|j;pD gIqkg;Qkwfh;jkG – & ,u7jk|hvpd;jk – # 0Pofh;pxt3pdlaopt]adG – & < – #. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. la’gdfwfh;jkG 16jcdo9eo;oF lv’9eo;o4h;o8k,.9F 9eo;o4h;omuj16jg[Nv’-hkp 9t,u7jk|hvpd;jk9eo;o4h;omuj16jg[Nv’0;kD 8q;1jk’G – * 16jg[Nv’-hkp0v’ – & faj’oAo – * < – &D * 16jg[Nv’0;k0v’ & faj’oAo * > &. zqogonjv’G { m5dM9eo;o4h;o]q[16jg[Nv’-hkp0v’ WF ltoAo m5dM9eo;o4h;o]q[9t,u7jk |hvpdj;k WD { m5dM9eo;o4h;o[;d16jg[Nv’0;k0v’ WF ltoAo m5dM9eo;o4h;o[;d9t,u7jk s^kpd;jk WD. $. [qfg/ydsaf. !D. dD 9eo;o4h;o8+wxoUF 9eo;o4h;o.fgxao9eo;o4h;o[;d s^n 9eo;o4h;o]q[ L – 7; – 21; 19; – 15; – 23; 15; 9; 8; 12; 2; 0; 3; – 1; – 6. 0D [aofk9eo;o4h;omuj.sh.o0=h Xd? 9eo;o4h;o.fgxaovq’xtdv[0v’ ]+ s^n ]– 8q;1jk’G + #∈]+ c]t − &∈]−. 13.

(18) @D. 9qj’8nj,g7njv’\kp ∈ s^n ∉ .lj[jvo9Eg,af.sh4nd8hv’D + # DDDD ]+ S + (DDDD ]+ S –!W) DDDD ]+ S + * DDDD ]+ S – 67 DDDD ]+ S − & DDDD ]– S –13 DDDD ]+ S + @$ DDDD ]+ S + !@_& DDDD ]+ ; + @(DDDD ]– S. #D. − &(DDDD ]– S. 9qj’[vdva[-yf0v’g,af H. − @W(DDDD ]– S. A; B; C; D; E; F c]t G. E. C. O. −4. −2. 0. A. I. −$. DDDD ]–S + ) DDDD ]– S − #(DDDD ]–S +!W$ DDDD ]–. D. F. B. G. D. 2. dD 9qj’c8h,cdo9eo;o4h;o 3fp;jk OI =1cm F O(0) c]t I(1)D 0D 9qj’;k’g,af A; B; C; D; E; F c]t G muj,uva[-yf + 5; –3; +7; +2; – 4; – 6; +8 8k,]efa[D 7D 9qj’;k’g,af A' , B' , D' c]t F' muj,uva[-yfgxao9eo;odq’dao0hk,da[va[-yf0v’ g,af A; B; D; c]t F 8k,]efa[D. &D. 9qj’[vdva[-yf0v’[aofkg,af]5j,oUD dD. C. 0D. N. 7D. D. B. A −10. −15. M. F. P. 103. M. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. $D. 108. D. E. C. −4. 3. 9qj’;k’ O(0) c]t I(1) 16j0=h 7D. 14. A.

(19) *D. .sh O(0)F I(1) c]t g,af9eo;o|bj’]5j,oUG M. A. D. I. O. F. B G. E. dD 9qj’[vdva[-yf0v’[aofkg,afgmy’oUD 0D 9qj’[vdlv’g,afmuj,uva[-yfdq’dao0hk,daoD 7D 9qj’[vdg,afg7yj’dk’0v’ [AB]; [DI]. ’D 9qj’;k’g,afg7yj’dk’0v’mjvo-nj8+wxoUrhv,ma’deoqfva[-yf0v’,aoG dD 9qj’xP[mP[9eo;o4h;o.oc8j]t76j8+wxoUG −( c]t + @S @$& c]t @&!S − _ c]t −!#S −( c]t +@S − _) &$@ c]t − _) #$@S ) c]t &S # c]t *S )! #(& c]t )! !_&S. #W( c]t − * !*_S − ( c]t + (S − _!$ c]t − _!$S @ $*) c]t @ &!& D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. (D. ສດິ. [AD]; [IE]; [MI].. 0D 9qj’9af]Po9eo;o4h;o.o0=h d 8k,]efa[0Bo X c8j|hvpsks^kp ?D )D. dD 9qj’8nj,g7njv’\kpG < ; >; = .lj[jvo9Eg,af.sh4nd8hv’D −!# DDDDD −!@S !! DDDDD !#S −!W DDDDD #S !W* DDDDD !*WS − $&#* DDDDD − $&#*S !& DDDDD −!(S −#& DDDDD − #&S −!$_ DDDDD − $!_S −)$@ DDDDD (@&S. −(#@. DDDDD −(#&S −_#* DDDDD −*#_S &(@! DDDDD− &(@!D. 0D 9qj’9af]Po9eo;o.o0=h d 8k,]efa[cI, X c8js^kpsk|hvp ?D _D. dD9qj’deoqf7jk0v’9eo;o4h;o a; b; c; d; e 8k,]efa[ c]t 16j]ts;jk’ − # sk + @ 0D 9qj’deoqf7jk0v’9eo;o4h;o x; y 3fp;jk −7 < x ≤ −6 c]t 25 > y > 23 7D 9qj’deoqf7jk0v’9eo;o4h;o m; n; p mujgxaova[-yf0v’g,af.ocdo9eo;o4h;o ]5j,oUG P. −16 −15. n. m. 15.

(20) !WD dD 9qj’0Po8nj,.lj8k8t]k’]5j,oUD –@. c,jo9eo;odq’dao0hk,. W –!_. c,jo9eo;odq’dao0hk,. +. –. ). +. (. & +#. 0D 9qj’la’gdfzqowfhIa[.oc4;mu ! c]t c4;mu # r;dohv’gsao;jkco;.fL gIqk4vowfh75o]adltot0v’9eo;odq’dao0hk,7nco;.fL. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. !!D dD 9qj’[vd9eo;o4h;o x muj16j]ts;jk’ – # c]t &D 0D 9qj’[vd9eo;o4h;o y 3fp;jk –!$ ≤ y < −12 D 7D 9qj’[vd9eo;o4h;o z 3fp;jk 8> z > 4. ’D 9qj’[vd9eo;o4h;o P 3fp;jk 0 < P ≤ 6 D. 16.

(21) [qfmu # dko[;d c]t dko]q[9eo;ome,t-kf dyf9tde dyf9tdemu ! dD 9qj’0ufvhv,7e8v[muj4nd8hv’D 7e8v[.shg]nvd. 0=h,6o s^n 7e4k, zqo[;d 4825 + 25196 gmqjk. A. B. C. 29911. 30021. 29121. 9. 1. 0. 9. 1. 0. wx16j[jvo9Eg,af0v’rqfmu|bj’ c,jo 2 . 6 − 79 = 137 8q;g]dmuj0kf wx16j[jvo9Eg,af0v’rqfmu|bj’ c,jo. 0D 9qj’fafcx’zqo0v’dko7yfw]jg]d]5j,oU.sh4nd8hv’D +. 2609 352 2591. +. 4725. −. 3907 7732. 2961 2609 0362. −. 8632 4725 4917. 7D 9qj’la’gdfzqowfhIa[9kd0=h 0 c]h;xtdv[xt3spd]5j,oU.sh4nd8hv’D .shlv’9eo;ome,t-kf8k,.9 a c]t b ; a > b F a − b c,jo9eo;omujgvqk,kDDDDDDDDX[;dF ]q[F 76oF sko? da[ b c]h;gmqjkda[ ltoAo le]a[9eo;ome,t-kf a , b c]t c g-yj’ a > b g,njv a − b = c c]h; a = DDDDDDDDDDDDDDDDD. 17. ສດິ. 4 . 1+ 27 = 518 8q;g]dmuj0kf. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !. a. D.

(22) dyf9tde @D mhk; mv’ |ad 47 kg F ok’ [5o,u |ad 45 kg c]t dtgxqJkg7njv’0v’mhk; mv’ |ad 8 kg D dD 4hkmhk; mv’ gxhdtgxqJkg7njv’0v’]k;0Bo-aj’F c]h;oE|ad];,mujxtdqf16j|hk9=-y’ -aj’faj’djk; c,jogmqjk.fL 0D ok’ [5o,u gxhdtgxqJkg7njv’0v’]k; c]h;0Bo-aj’F oE|ad];,xtdqf16j9=0v’-y’-aj’ c,jo 50 kg D 9qj’7yfw]joE|addtgxJqkg7njv’0v’ ok’ [5o,uD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. dyf9tde #D lk,vhkpohv’G mhk; 7erk ,u\kd[uj 38 |j;pF mhk; 7erPo ,u\kd[uj 35 |j;p c]t ok’ ,vo]u ,u\kd[ju|hvpd;jkmhk; 7erPo 7 |j;pD dD 9qj’7yfw]j9eo;o\kd[uj0v’ok’,vo]uD 0D grnjv7yfw]j9eo;o\kd[juma’\qf0v’lk,vhkpohv’gryjogIafco;.fL 7D 9qj’xtdv[xt3spd]5j,oU.sh4nd8hv’D • dko[;d c,jodko7eo;ogrnjv7yfw]j DDDDDDDDDDDDDDDDDD 0v’lv’9eo;omuj,usq;|j;p vaofP;daog0qhkdaogxao9eo;ofP;D • dko]q[ c,jodko7eo;ogrnjv7yfw]j DDDDDDDDDDDDDDDDDDD 0v’lv’9eo;omuj,usq;|j;p vaofP;dao3fp;jk 9eo;omu|bj’8hv’s^kpd;jk s^n gmqjkda[9eo;omulv’D. @D .97;k, dD dko[;d9eo;ome,t-kf dko[;d9eo;ome,t-kf c,jodko7eo;omuj7yfw]jzqo[;d Xs^n 7yfw]jdko3I, gvqklv’ s^n s^kp9eo;o ? g0Qkgxao9eo;ofP;D laopt]adfh;pg7njv’\kp < + < 8q;1jk’G 2609 + 352 =. 2961. rqf0v’zqo[;d zqo[;d. • 2961 gvUo;jk<zqo[;d0v’ 2609 da[ 352 <. • 2609 c]t 352 gvUo;jk<rqf<. 18. 0v’zqo[;d.

(23) 75o]adltot • zqo[;d0v’lv’ s^n s^kp9eo;ome,t-kfc,jo9eo;ome,t-kfD \kp7;k,;jkG .sh a ∈` c]t b∈` gIqkwfh a + b ∈` s^n g;Qkvudco;|bj’;jkGdko[;d.odj5,9eo;ome,t-kf gxaodqfdko7eo;orkp.oD •. zqo[;d0v’lv’9eo;ome,t-kf[+xjPocx’ 4hk;jkgIqkla[xjPo[jvorqf8jk’MD \kp7;k,;jkG .sh a ∈` c]t b∈` gIqkwfh a + b = b + a 8q;1jk’G. 7 + 5 = 5 + 7 = 12. s^n g;Qkvudco;|bj’;jkG dko[;d.odj5,9eo;ome,t-kf ,u75o]adltotla[xjPo [jvowfhD zqo[;d0v’s^kp9eo;ome,t-kf[+xjPocx’F 4hkgIqk3I,gvqk[k’rqf0v’r;d ,aog0Qkdaodjvo c]h;cmofh;pzqo[;d0v’r;d,aoD \kp7;k,;jkG .sh a ∈` , b∈` c]t c∈` gIqkwfh a + b + c = ( a + b ) + c = a + ( b + c ) s^n g;Qkvudco;|bj’;jkG dko[;d.odj5,9eo;ome,t-kf ,u75o]adltot3I,\6jwfhD 8q;1jk’G ( 7 + 5) + 8. 7 + ( 5 + 8) 7 + 13 20. 12 + 8 20. faj’oAo. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. •. 7 + 5 + 8 = ( 7 + 5 ) + 8 = 7 + ( 5 + 8 ) = 20. g8adoyddko[;d • djvovnjo oadIPo8hv’[;dlv’9eo;omuj,u|bj’8q;g]dwfh1jk’vaf8t3o,af c]t xklt9kdg7njv’7yfw]jD 8q;1jk’G 7 + 5 = 12 ; 9 + 9 = 18 ; 6 + 8 = 14 . 19.

(24) •. +. zqo[;d0v’lv’9eo;omuj0Pofh;ps^kp8q;g]dD 8q;1jk’G dko7yfw]j. 2+19+17 +16 5 8 4 9 8 8 2 5. 6 + 9 = 15. 70 + 40 = 110 900 + 800 = 1700 2000 + 5000 = 7 000 2976 + 5849 = 8875. 0D dko]q[9eo;ome,t-kf dko]q[ g-yj’laopt]ad fh;pg7njv’\kp< − < c,jodko7eo;omuj7yf w]j7;k,c8d 8jk’ Xzqo]q[? 0v’lv’9eo;oD • lv’9eo;ome,t-kf8k,.9 a c]t b 3fp;jk a > b S a − b c,jo9eo;omuj gvqk,k[;dda[ b c]h;gmqjkda[ a D 4hk a − b = c c]h; b + c = a D 8q;1jk’G • 3907 gvUo;jk zqo]q[0v’ 8632 da[ 4725 D 8632 − 4725 = 3907 • 9eo;o 8632 c]t 4725 gvUo;jk rqf 0v’ zqo]q[D rqf0v’zqo]q[ zqo]q[ • 8632 gvUo;jk 8q;8ah’]q[F 4725 gvUo;jk 8q; ]q[D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. •. 75o]adltot • zqo]q[lv’9eo;ome,t-kf3fpgvqk9eo;os^kpd;jk ]q[9eo;o|hvpd;jk c,jo9eo;ome,t-kfD \kp7;k,;jkG .sh a ∈` c]t b∈ ` , a > b gIqkwfh a − b ∈` • zqo]q[lv’9eo;ome,t-kfF ]efa[rqf,u7;k,le7aoF gIqk[+lk,kfla[xjPo [jvowfhD \kp7;k,;jkG .sh a ∈` ; b∈` F gIqkwfh a − b [+gmqjk b − a . laopt]ad a − b ≠ b − a F 8q;1jk’G 9 − 5 ≠ 5 − 9 grkt;jk 9 − 5 = 4 S c8j;jk 5 − 9 [=jlk,kf7yfw]jwfh.od5j, ` grkt;jk 5 < 9 D 20.

(25) g8adoyddko]q[G d;f7noG −. 8 2 9 5 4 +15 8 +18 3 7 0 7. +. 4588 37 0 7 8 295. #D [qfg/ydsaf !D. 9qj’7yfw]jzqo[;d c]t zqo]q[]j5,oU X[+.-hg7njv’7yfw]j?D dD 439 + 257 S 0D 1986 + 734 S ’D 7849 + 251 + 9090 S 9D 15 + 2895 + 357 S -D 357 − 153 S pD 4910 − 254 S 4D 47201 − 28795 S 8D 13548 − 8937 S. 7D 984 + 3976 S lD 4075 + 318 + 1482 fD 4583 − 956 S mD 107 − 59 D. 9qj’oe.-h75o]adltotdkola[xjPo[jvowfh c]t dko3I,\6jwfh 0v’dko[;d grjnv,k 7yfw]jg]d]5j,oU.sh’jkp0bhoD dD 24 + 13 + 16 + 17 S 0D 25 + 39 + 75 + 85 + 61 S 7D 438 + 247 + 192 + 62 + 153 S ’D 131 + 547 + 53 + 70 D. #D. 9qj’gvqk8q;g]d8nj,.lj[jvo9Eg,af grnjv.shdko7yfw]jgxaowxwfh c]t 4nd8hv’D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. @D. +. −. 438 . 5. 7.9 . 5 3. . . 222. +. .27 10 3 7. .. −. . 4 9 15 . 3 . 6. +. .. 8 54. 973. −. +. 6. . . 87 16 4. 21. . 659 8. . 2.63. −. 16 0. + 7.5 + 2.99 .4 41. . 6 91 1. 2 . 6 6 . 6. −. 387 2 . + 90 . 4 + ... 70 64 6 3 9 30. 2 . 386 7. 7..

(26) $D. 9qj’gvqk8q;g]d8nj,.lj[jvo9Eg,afgrnjv.shdko7yfw]jgxaowxwfh c]t 4nd8hv’D dD .386 + 45 . + 6 .97 = .16 . 2 S 0D 38. + 6.3 + .85 = 1547 7D .362 + 45.0 + 80. = 6.59 S ’D 997. + 3.96 + .537 = .04.6 9D ..79 + 90.2 + 63. = .4428 S lD .987 − ..3 = 633. -D .38. − 2.6 = 5.08 S pD .367 − ..9 = 388. S fD .34.0 − 68.98 = 2.61 S 8D .38. − 42.4 = 2.91 S. &D. 9qj’-vdsklv’9eo;ome,t-kf]Po4afdaoF I6h;jkzqo[;d0v’r;d,aogmqjk 1275 D. *D. 9qj’-vdsklk,9eo;ome,t-kf]Po4afdao x + y +z = 435 D. (D. xB,sq;|bj’,u 146 |hkD 4k,;jkgrnjv0PooEg[uh|hk0v’xB,Ivf|hkmu 146 gryjo8hv’ 0Po8q;g]d9adgmjnvL XI6h;jk0Po|bj’gmnjvwfh|bj’8q;g]d?D. )D. gryjo0PooEg[u|hk0v’xB,sq;|bj’3fpg]uj,9kd|hk 1, 2,3, 4,..., gryjowfh0Po8q;g]d 8jk’M 516 gmnjvD 4k,;jkxB,sq;faj’djk;,u9ad|hkL XI6h;jk0Po|bj’gmnjvwfh|bj’8q;g]d?D. _D. lt|k,dy]kcsj’|bj’[ao95wfh 10 000 7qoF 4hk,uz6hg0Qk-q,dy]k 7875 7qoD 4k,;jk ,u9ad[jvooaj’mujgxqjks;jk’L. c]t. z. F I6h;jk. x < y <z. c]t. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. x, y. !WD 3I’IPo,afmtpq,csj’|bj’,uoadIPoma’\qf 1522 7qoD .ooU 0Ao ,D! ,u 439 7qoF 0Ao ,D@ ,u 413 7qo c]t 0Ao ,D$ ,u 295 7qoD 4k,;jkoadIPo0Ao ,D# ,u9ad7qoL !!D xB,c[[IPo7toyflkf |bj’sq; c]t xB,c[[IPorklk]k;lv’sq; ]k7k];, dao17 000 du[D xB,c[[IPo7toyflkf lv’sq; c]t xB,c[[IPorklk]k;lk, sq; ]k7k];,dao 29 000 du[D dD 4k,;jk xB,c[[IPo7toyflkf @ sq; c]t xB,rklk]k; 4 sq; ,u]k7k];,dao gmqjk.fL 0D 4k,;jk xB,c8j]t-toyf,u]k7ksq;]tgmqjk.fL. 22.

(27) !@D mhk; 7es^hk ,ug’yo 3550 du[F r+0v’]k;gvqk8jn,.shvud 4850 du[D mhk; 7es^hk [vd;jk .og;]koU]k;,ug’uos^kpd;jk ok’ pvf2hk gvNvp0v’]k; 1700 du[D 4k,;jk ok’ pvf2hk ,ug’yogmqjk.fL !#D ok’ mjP’me wfhltlq,ltc8,ma’\qf 1253 vaoD .ooU,u 357 vao c,joltc8,gxao I6[laf8jk’MD 4k,;jk ltc8,0v’ ok’ mjP’me muj[+c,joI6[laf ,u9advaoL !$D 4a’|j;p|bj’[ao95oEwfhs^kpd;jk75 3 ]yf c]t |hvpd;jkws 15 ]yfD .sh I6h;jk ws[ao95oEwfh 25 ]yfD 4k,;jk 4a’|j;poAo[ao95oEwfh9ad]yfL c]t 75[ao95wfh9ad]yfL. ສດິ. !&D 3I’IPo,afmtpq,csj’|bj’F 0Ao ,D! ,ulv’shv’7nG shv’,!Dd c]t shv’,!D0F shv’ ,!Dd ,uoadIPopy’ 23 7qo c]t oadIPo-kp 20 7qoD shv’,!D0 ,uoadIPoma’\qf 39 7qoD .olv’shv’0Ao ,D! ,uoadIPopy’ma’\qf 41 7qoD dD 4k,;jk 9eo;ooadIPoma’\qf0v’0Ao ,D! ,u9ad7qoL 0D 4k,;jk 9eo;ooadIPopy’.oshv’,!D0 ,u9ad7qoL. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !*D mhk; 7volts;ao wfh9af’ko;aogduf0v’]k;7q[Iv[12 xu .o,N;aomu 15 85]kF r+0v’ ]k;wfhg;Qkda[]k;;jk xuoUg9Qk,uvkp5|hvpd;jkvhkpdqd0v’g9Qk 25 xur=fuD 4k,;jk xu oUvhkpdqd0v’ mhk; 7volts;ao ,uvkp59adxuL !(D ok’ ;aofu wx8ts^kfF ]k;,ug’yo16j.odtgxqJk .[ 50 000 du[ |bj’.[ c]t .[ 5000 du[ lk,.[D g,njvda[,kIvfgInvo g’yo0v’]k;pa’gs^nv16j.odtgxqJk ,u.[10 000 du[ |bj’ .[F .[ 2 000 du[ ,uluj.[ c]t .[1000 du[ ,ulk,.[D 4k,;jk ]k;9jkpg’yo16j8ts^kf ma’\qfgmqjk.fL !)D ]qfg,7ao|bj’ ,uz6h3fplko 40 7qo vvd9kd8ts^kfg-QkD g,njvwxIvfxhkp9vf[jvo|bj’ z6h3fplkowfh]q’]qf9eo;o|bj’ c]t 0Bovud 20 7qoD djvovvd9kdxhkp gryjooa[z6h3fp lko.o]qf,u 36 7qoD 4k,;jk z6h3fplko]q’]qf9ad7qoL. 23.

(28) !_D gryjo.-h8q;g]d 1,3 c]t. 5. grnjv0Po9eo;omuj,ulk,8q;g]d8jk’daoD 3 5 gIqkwfh 135. 1 5. 3 5. dD 9qj’xtdv[Ijk’’jkw,hgmy’oU.sh7q[4h;oD 0D 4k,;jk gryjolk,kf0Powfh9ad9eo;o L 7D 9qj’9af]Po]efa[9eo;owfh,kc8j|hvpsks^kpD ’D 9qj’7yfw]jzqo[;d9eo;omuwfh,kma’\qfD 9D 9qj’7yfw]j7;k,s^5f]jno]ts;jk’9eo;os^kpl5f c]t 9eo;o|hvpl5fD @WD dD 9qj’xtdv[ c]t 7yfw]jdko7eo;o]5j,oUD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 S = 10 + 9 + 8 + ............................ + 1 S +S = 11 + 11 + ................................ + 11 2S = 11× .................. S = .......................... 0D oe.-hdko7yfw]j.o0=h Xd? grnjv7yfw]jzqo[;d]5j,oUG a. S = 1+ 2 + 3+ 4 + 5 +...+ 99 b. S = 1+ 2 + 3+ 4 + 5 +...+ 200. @!D 9qj’xtdv[8k8t]k’]5j,oU.sh7q[4h;o c]t 4nd8hv’ 3fp.shzqo[;d8k,4aoF c4; c]t 8k,glAogoaj’9v,gmqjkdaoD dD 0D 2. 7. 6. 18 13. 3. 8. 24. 15. 17.

(29) [qfmu $ dko[;d c]t dko]q[9eo;o4h;o !D dyf9tde dyf9tdemu ! mhk; 7e pjk’8k,cdo9eo;o4h;o d fh;p[kfdhk;lt\jeltg\umuj,u7;k,d;hk’ OA g-yj’ O(0), A(1) c]t g]uj,8Qo9kdg,af O.. d C' B' A' O A B C. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. dD 4hkmhk; 7e pjk’wx|hk & [kfdhk;9kd O c]h;wx|hk8nj,vud # [kfdhk;F gsao;jk mhk; 7e wxg4y’95f HX)?D ltcf’;jk w]pt mk’wxwfh ) sq;|j;p s^n (+5) + (+3) = +8 D 4hk]k; 4vpsâ’9kd O luj [kfdhk; c]t 4vpsâ’vud shk [kfdhk; c]h;9twx16j J' ( −8 ) D ltcf’;jk (−4) + (−5) = −9 D 9qj’xtdv[8k8t]k’]5j,oUD dkog7njvomuj9kd. muj8A’l5fmhkpmujwxg4y’mP[.lj ltcf’dkog7njvo g,af O mujfh;p9eo;o wx|hk & [kfdhk; 8k,fh;p wxg4y’g,af HF ltcf’;jkwx (+5) + (+3) = +8 wx|hk8nj,vud # [kfdhk;D |hk ) sq;|j;psjk’9kd O D 4vpsâ’ $ [kfdhk; c]t wxg4y’g,af J′F ltcf’;jk4vp (−4) + (−5) = −9 4vpsâ’vud & [kfdhk;D sâ’ _ sq;|j;psjk’9kd O D O. wx|hk @ [kfdhk; 8k,fh;p wx|hk8nj,vud * [kfdhk;D 4vpsâ’ # [kfdhk; c]t 4vpsâ’vud ( [kfdhk;D 25.

(30) wx|hk _ [kfdhk; c]t 4vpsâ’vud $ [kfdhk;D wx|hk # [kfdhk; c]t 4vpsâ’vud & [kfdhk;D 4vpsâ’ # [kfdhk; 8k, fh;pwx|hk _ [kfdhk;D 4vpsâ’ ) [kfdhk; c]t 8k,fh;pwx|hk $ [kfdhk;D 7D 3fpla’gdf9kdzqowfhIa[.o4aomulk,0v’8k8t]k’gmy’oUF 9qj’4vogvqksâddko 0v’dko[;d9eo;o4h;o .od=]tou8jk’MD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. dyf9tdemu @ g-ajofP;daoda[16j.o ` F .o ] 4hk a− b = c c]h; c c,jo9eo;o4h;omuj[;dda[ b gmqjkda[ a \kp7;k,;jk b+ c = a. 8q;1jk’G dD (+3) + (+1) = +4 ltcf’;jk (+4) − (+3) = +1 s^n (+4) − (+1) = +3 0D (−3) + (+5) = +2 ltcf’;jk (+2) − (−3) = +5 s^n (+2) − (+5) = −3 9qj’xtdv[8k8t]k’]5j,oU8k,8q;1jk’gmy’oUD a. b. +4. +3. +4. +1. +2. +5. +2. −3. +7. −4. −5. +8. −9. +6. −4. −4. −3. −7. a−b=. (+4) − (+3) = +1. 26. a+. X−b? =. (+4) + (−3) = +1.

(31) −8. −4. −12. −9. 9qj’xP[mP[zqowfhIa[.o4aomulk, c]t luj c]h;4vogvqksâddko0v’dko]q[9eo;o 4h;oD. @D .97;k, !D w]ptsjk’9kd 0 D .sh A; B; C; D gmy’cdo9eo;o4h;o B −6. C. O ( 0 ) , I (1) :. O. −3. I. A. D. +4. +6. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. g,af D ( +6 ) F faj’oAo ];’pk;0v’mjvo-nj [OD] c,jo OD = 6 gryjogvUow]ptsjk’ 0v’9e o;o4h;o +6 9kd W c,jo *D g,af C ( −3) F faj’oAo w]ptsjk’0v’9eo;o 4h;o –3 9kd W c,jo];’pk;0v’ mjvo-nj [OC] c,jo OC = 3 D 4hk g,af M ,uva[-yf x, w]ptsjk’0v’9eo;o4h;o x 9kd W c,jo];’pk;0v’ mjvo-nj [OM] D w]ptsjk’9kd W 0v’9eo;o4h;o8jk’Mgxao9eo;o[;dg]NvpMD. @D dko[;d9eo;o4h;oD grnjv7yfw]jzqo[;d0v’lv’9eo;o4h;omuj,ug7njv’\kp7ndaoF gryjoxt8y[affaj’oUG [;dw]ptsjk’9kd W 0v’r;d,ao c]h;gvqkg7njv’\kp7ndao0v’r;d,ao.lj8+|hk 0v’zqo[;dD 8q;1jk’G dD (+3) + (+7) = +10 S grkt;jk +3 ,uw]ptsjk’9kd W gmqjk # c]t +7 ,uw]ptsjk’ 9kd W gmqjk ( c]t 3 + 7 = 10 D 0D (−8) + (−12) = −20 S grkt;jk − 8 ,uw]ptsjk’9kd W gmqjk ) c]t − 12 ,uw]pt sjk’9kd W gmqjk !@ c]t 8 + 12 = 20 D 27.

(32) grnjv7yfw]jzqo[;d0v’lv’9eo;o4h;omuj,ug7njv’\kp8jk’daoF gryjoxt8y[affaj’oUG gvqkw]ptsjk’0v’9eo;omujsjk’9kd W s^kpd;jk ]q[w]ptsjk’0v’9eo;omujsjk’ 9kd W |hvpd;jk c]h;gvqkg7njv’\kp0v’9eo;omujsjk’9kd W s^kpd;jk.ljD 8q;1jk’G dD (−6) + (+8) = +2 S grkt;jk 8 − 6 = 2 c]t g7njv’\kpc,jog7njv’\kp0v’ + ) grkt + ) 16jsjk’9kd W s^kpd;jk − *D 0D (−7) + (+4) = −3 S grkt;jk 7 − 4 = 3 c]t g7njv’\kpc,jog7njv’\kp0v’ −7 grkt −7 16jsjk’9kd W s^kpd;jk + 4 D 75o]adltotD dko[;d.odj5,9eo;o4h;o gxaodqfdko7eo;orkp.oD dko[;d.od5j,9eo;o4h;o ,u75o]adltotla[xjPo[jvowfh c]t 3I,\6jwfhD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 8q;1jk’G dD (−7) + (+4) = (+4) + (−7) = −3 Xla[xjPo[jvo? 0D (+8) + (−6) + (−5) = [(+8) + (−6)] + (−5) = +8 + [(−6) + (−5)] = −3 X3I,\j6? dko[;ds^kp9eo;o4h;o7yfw]jwfhfh;plv’;ymufaj’oUG ;ymumu !G gryjo7yfw]jzqo[;dgmnjv]tlv’rqfc8j-hkpsk0;k9qog4y’rqfl5fmhkpD 8q;1jk’G A = (+7) + (+4) + (−3) + (−4) + (5) A = +11 + (−3) + ( −4) + ( +5) A = +8 + ( −4) + ( +5) A = +4 + ( +5) A=9. ;ymumu @G [;d9eo;omuj,ug7njv’\kp7ndaodjvo 9kdoAo9bj’[;dzqo[;dmujwfh,k.ljdaoD 8q;1jk’G A = (+7) + (+4) + (–3) + (–4) + (+5) + (–2) A = (+ 7) + (+4) +(+5) + (–3) + (–4) + (–2) A = (+ 16) + (–9) A=7. 28.

(33) zqo[;dlv’9eo;odq’dao0hk,daogmqjkl6og]NvpMD xUo7noG4hkzqo[;dlv’9eo;ogmqjkl6oc,jolv’9eo;ooAogxao9eo;odq’dao0hk,daoD. 8q;1jk’G dD X–5) + (+5) = 0 0D 7 + x = 0 gIqkwfh x c,jo9eo;odq’dao0hk,0v’ 7 F faj’oAo x = –7. #D dko]q[9eo;o4h;oD gIqkI6hc]h;;jkG 9eo;odq’dao0hk,0v’ x c,jo −x grnjv7yfw]jzqo]q[0v’lv’9eo;o4h;oF gryjo7yfw]jzqo[;d8q;8A’]q[da[9eo;odq’dao 0hk,0v’8q;]q[D .shlv’9eo;o4h;o8k,.9 a c]t b ; a − b = a + (−b). 8q;1jk’G dD 7 − 5 =7 + (−5) = 2 S 0D 3 − (−5) =3 + 5 = 8 S 9D −13− 9 = −13 + (−9) = −22.. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 7D −3 − (−9) = −3 + 9 = 6 S ’D −12 − (−5) = −12 + 5 = −7. $D dko7yfw]j\;f7eo;o0v’9eo;o4h;oD dD \;f7eo;omuj[+,u;q’g]a[D \;f7eo;omuj[+,u;q’g]a[lk,kf7yfw]jwfhfh;plv’;ymufaj’oUD. 8q;1jk’G A = −14 − (−8) + 15 − (−7) + (−8) −9. ;ymumu!G 0Po\;f7eo;o A gxaozqo[;d c]h;7yfw]j7nda[dko[;ds^kp9eo;o4h;o. ;ymumu @G. A = −14 − (−8) + (+15) − (−7) + (−8) −(+9). A = −14 + 8 + 15 +7 + (−8) +(−9). A = −14 + 15 +7 +(−9). A = −14 +(−9) + 15+7 A = −23 +22 A = −1 0Po\;f7eo;o A gxaoI6[0Posp=hc]h;3I,gvqkrqfmuj,ug7njv’\kp16j|hk7n. dao,kw;hoedaoF 7yfw]j7nda[dko[;ds^kp9eo;o4h;oD 8q;1jk’G. A = −14 − (−8) + 15 − (−7) + (−8) −9 A = −14+ 8+ 15+7 + (−8) +(−9). A = 30+(−31) A = −1 29.

(34) 0D \;f7eo;omuj,u;q’g]a[D \;f7eo;omuj,u;q’g]a[F gIqk7yfw]j.o;q’g]a[djvoD 8q;1jk’G A = (− X!+)−!@? +X−_ + !!? A = (− X−#? +X+@? A = (+#+@ A = !@. #D [qfg/ydsaf 9qj’7yfw]jg]d]5j,oUG. 2D. 3D. B = (−11) + (+12); D = (+4) + (+15); F = (−18) + (+7); H = (−18) + (+15); J = (−6) + (+6); L = −23 + 28; N = −17 + (−21); S = (−601) + (−11). ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. A = (−6) + (+11); C = (+12) + (−5); E = (−6) + (−11); G = (−15) + (−12); I = (−19) + (−18); K = (+18) + (−24); M = 53 + 76; P = (−127) + (+161);. ສດິ. !D. 9qj’7yfw]jg]d]5j,oUG A = X−16) − (+11); C = X−6) − (−15); E = X−12) − (−12); G = X+7) − (−18); I = X+16) − (−71); K = X−167) − (+311);. B = X+12) − (−5);. D = X−14) − (+21); F = X+13) − (+15);. H = X+25) − (+11); J = X+96) − (−101); L = X−9) −(+21);. 9qj’8nj,9eo;omuj4nd8hv’.lj[jvo9Eg,afD dD DDD+ (+7) = −11 7D (−3) + DDD = −6 9D DDD − (+7) = −12 -D DDD − (−15) = +27 fD 7 + DDD = −3 4D 18 − DDD = 11. 0D DDD + (−12) = 15 ’D (−18)+ DDD = 2 lD DDD − (−3) = −6 pD DDD − (+16) = 18 8D (−13) − DDD = −5 mD 3 − DDD = 8 30.

(35) $D. 9qj’7yfw]j\;f7eo;o]5j,oUG A = (−18) +(−15)+ (+12)+ (−7)+ (+15); B = (−26) +(−4)+ (+17)+ (−3)+ (−15) C = (+16) +(+11)+ (−5)+ (−13)+ (+2) D = (−3) −(+12)+ (+18)− (−7)+ (+17) E = 12 − (−15)+ (−6)− (−9)− (+12) F = (−5) −(−6)+ 16− (+12) −15 G = (−18) −[(−15)+ (+12)+ (−7)]+ (+15) H = (−18) +[(−15)− (+12)]−[−7+ 15] I = [(−18) +(−15)]−[(+12)+ (−7)]+ 35 J = (−18) −[(−15)+ (+12)]+ [(−47)+ (−15)]. *D. 0D ’D lD pD 8D. (−8)− x =2; x+ (+14) = −5);. (+13) −x= −1 x+(−9) =18;. 3+(x−6) = 3−(−4);. 9qj’8nj,8k8t]k’]5j,oU.sh4nd8hv’D a) –6 x y x+y 8. 5 –7 12. 4. –11. –3 –9. –2 7. –16. b) x y x−y. −12 17. 12 –9 5. −16. 8 –7. 3. –4 20 –10. 4 0 1. 5 –5 –1. c) x y z x+y+z. 1 –3 5. 3 –1 4. 5. 31. ສດິ. 9qj’-vd7jk0v’ x ]5j,oUG dD x + (−7) =15; 7D x −(−9) = −4; 9D (−15) −x= 8; -D x− (+3) = −15; fD (19−3) + x=(25−18);. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. &D. –5 0. 7 9 –1.

(36) (D. 9qj’8nj,8k8t]k’]5j,oUfh;p !* 9eo;o4h;oc8j −( sk ) 3fp;jk9eo;o|bj’.-h wfh7A’fP; c]t zqo[;d 0v’9eo;o16j8k,c8j]t4aoF c8j]tc4; c]t c8j]t glAogoaj’9v, gmqjkdaoD. 6. –7. 4. –4. –2. 7. 5. 3 –3. )D. 9qj’xtdv[I6[fk;]5j,oU3fp.shzqo[;d8k,c8j]tc4;gmqjkda[ @D 6 −4. 7. −3 0. _D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 4. ສດິ. −7. 9kdI6[]5j,oUF gryjog7njvomjuwx8k,mk’]6dlvo3fpdko[;d.sh9eo;omujrq[D +8. −3. −6. −1. + 2 = .......... +12. −9. −1. dD ,u9admk’g]nvdg,njvg7njvomuj9kd −1 wxsk +2D 0D 9qj’deoqfglAomk’ g-yj’zqo[;d9eo;omjurq[ma’\qfgmqjkda[l6oD 7D glAomk’.fmuj.shzqo[;d9eo;omujrq[s^kpl5fD ’D glAomk’.fmuj.shzqo[;d9eo;omujrq[|hvpl5fD 9D ,uglAomk’mujuzqo[;d9eo;omujrq[gmqjklk, s^n[+L 4hk,u 9qj’[vdglAomk’oAoD. 32.

(37) [qfmu & dko76o c]t dkosko9eo;ome,t-kf !D dyf9tde dyf9tdemu ! !D mhk; lk]u -NxB,0Posqdsq;M]t !&WW du[D g9Qk0v’Ihko7yfw]j]k7kxB,0Posqdsq; fh;pdko[;d !&WW -Edaosqdgmnjvfaj’oUG !&WW + !&WW + !&WW + !&WW + !&WW + !&WW = _WWW dD 4k,;jkovd9kd;ymuoUgryjolk,kf7yfw]j]k7kxB,sqdsq;wfhfh;p;ymu.fvudL 0D !&WW + !&WW + !&WW + !&WW + !&WW + !&WW = _WWW gmnjv0v’ 1500 = 9000 ×1500 = 9000 D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 7D 9qj’4vogvqkdkorq;rao]ts;jk’dko[;d c]t dko76o3fpdkoxtdv[xt3spd ]5j,oUG dko76o !&WW da[ * c,jodko[;d DDDDDDDDDDDDDDD -Edao DDDDDDDDD gmnjvD. @D xB,7toyflkf ,D! lk,sq;]k7k];,dao @!WWW du[D 9qj’7yfw]j]k7kxB,7toyflkf faj’djk; 9eo;oly[shksq;F ly[sqdsq; c]t -k;sq;fh;ps^kp;ymu8jk’daoD dyf9tdemu @ gryjowfhc8h,I6[ltcf’dkocpd @$ vvdgxaozqo76o0v’lv’9eo;ome,t-kffaj’oUG ) + ) )+. 3×8= 24. 3 + 3+ 3 + 3 + 3 + 3 + 3 + 3 8×3. dD 9kdzqodko7yfw]jwfh,k ltcf’.shgIqkgsao;jkdko76o0v’lv’9eo;ome,t-kf ,u75o]adltotco;.fL. 33.

(38) 0D 3fpoe.-h;ymufP;da[0=h dD 9qj’8nj,.lj[jvo9Eg,af.sh4nd8hv’D 2x4 + 2x4 + 2x4. …..x…..= 24. (3 x 2) + (3 x 2) + (3 x 2) + (3 x 2) ….x…..= 24. 108 9 .... ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ... × 9 = 108 ... = 108 ÷ 9. ສດິ. dyf9tde # mhk; ]ugIu c]t ok’ [5o rhv,da[\6j];,ma’\qf _ 7qo c[j’xao\kd[uj !W) |j;p g-yj’ m5d7qo8hv’wfhgmqjkdaoD mhk; ]ugIu wfhgvqk\kd[juma’\qf,k1kpgxaogdQkdv’3fpgvqk \kd[ju8nj,.ljc8j]tdv’gmnjv]t|j;pF ok’ [5o g;Qk;jkc[j’c[[oU,aopkd c]t glpg; ]kS r;dgIqkrhv,dao7yfw]jdjvo c]h;9bj’c[j’dao8k,zqodko7yfw]j faj’oUG. 9eo;omuj4nd8hv’16j[jvo9Eg,af c,jozqo0v’dkoc[j’xao s^n zqosko 108 .sh 9 D 9qj’1A’1nodko7yfw]j7no;jk4nd8hv’fh;pdko]q[ !W) vvdgmnjv]t _ 8yf8+dao9qowfh zqoIa[gmqjkl6oF g-ajoG !W) − _ = 99S __ − _ = _WS _W − _ = )!S DDD 4k,;jkG gIqk]q[8yf8+dao9adgmnjvF zqo]q[9bj’gmqjkl6oL gIqk4vowfhdkorq;rao]ts;jk’dko]q[ c]t dkoskoco;.fL. @D .97;k, !D dko76o9eo;ome,t-kfD .sh9eo;ome,t-kf a c]t n S dko76o a da[ n laopt]adG n× a c,jodko7yfw]j0v’dko[;d a -Edao n gmnjvD \kp7;k,;jkG a + a + a + a + … + a = n× a. n gmnjv 34.

(39) laopt]adG n ×a vjko;jk n 76oda[ a F zqowfhIa[gvUo;jk zqo76o n da[ a. lj;o;jk n c]t a mujgvqk,k76odaogvUo;jk lj;o76oD dko76oc,jodko7eo;omuj.-hcmodko[;d9eo;o.f|bj’muj-EdaoD 8q;1jk’G 138 24 552 276. ×. gIqkwfhG !#) × @$ = ##!@. 3312. s^n. lj;o76o. 138 + 138 + 138 + ... + 138 = 3312 24 gmnjv. zqo76o. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. @D 75o]adltot dko76ola[xjPo[jvowfh \kp7;k,;jk le]a[m5dM9eo;ome,t-kf a c]t b gIqkwfh a × b = b × a dko76o3I,\j6wfh \kp7;k,;jk le]a[m5dM9eo;ome,t-kf a; b c]t c gIqkwfh Xa×b?×c = a×(b×c) = a×b×c. .sh9eo;ome,t-kf a 8k,.9S gIqkwfh. a × ! = a.. 8q;1jk’G dD )×( = (×) = 56 0D (2× (?×# = !$ ×# = $@ 2×X(×#? = @×@! = $@ 7D. ( 2 × 7) ×3 = 2 ×(7×3) = 2 × 7 × 3 = 42. 7 ×1 = 7. #D dkosko9eo;ome,t-kf .sh9eo;ome,t-kf a c]t b 8k,.9 3fp;jk a > bF zqosko a .sh b laopt]ad a ÷ b c,jodko7yfw]jcmodko]q[ b vvd9kd a 8yf8+dao9qozqo]q[gmqjk r 3fp;jk r < b. 35.

(40) a − b − b − … − b = rF. g-yj’ r < b. ltcf’;jk. q gmnjv. a b r q. gryjogvUo. q. ;jk zqosko c]t r ;jk 8q;glf 0v’ a sko.sh b D. 8q;1jk’G. 23 − 7= 16; 16 − 7= 9; 9 − 7= 2. s^n. c,jo @# ( @ # # c,jozqpsko c]t @ c,jo38glfD. 23 − 7− 7 − 7 = 2 < 7 D 3gmnjv. dkoskoc,jodko7eo;omuj.-hgrnjv-vdskzqosko c]t 8q;glfD !D. 9qj’8A’[A’g]dc]h;7yfw]jzqo76o]5j,oUD dD !(& × !&S 9D *() × *_S lD )(_ × )(S 0D @$)# × _&S 7D _* @@( × !&*S -D _*_ × _(*S ’D (&# × *!)S pD *#& × ! WWWS. @D. #D. 9qj’7yfw]jzqo76o]5j,oU X[+.-hg7njv’7yfw]j?D 7D @& × !@ × )S dD X!@ × &? × (S 0D *W × X!W × (?S ’D (× _ × & × !WS. fD # (W! × !WWS 8D *W& × !W × !WWS 4D $ WW$ × !WW × ! WWW × !WS mD $!( × ! WWW × !WW × !WD. ສດິ. [qfg/ydsafD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. #. 9D X) × $ × &? × *S lD X!& × *? × X@& × $ × )?D. 3fpoe.-hdkorq;rao]ts;jk’dko76o c]t dkosko 9eo;ome,t-kf aF b c]t c, a = b × c gIqkwfh a ÷ b = c s^n a ÷ c = b. 9qj’-vdsk7jk0v’ x ]5j,oUD lD *W ÷ x = &S dD * × x = $@S -D x ÷ @& = $S 0D x × @! = @!WS pD x ÷ !! = !$S 7D !@ × x = $)S fD !@* ÷ x = !)S ’D #& × x = !W&S 4D !#* ÷ x = !(D 9D & × x = (&S. 36.

(41) 9qj’cmo[jvo9Eg,affh;p8q;g]dF grnjv.shdko7yfw]jzqo76o c]t zqosko]5j,oUgxao9y’D dD 0D 7D ’D ×. 691. ••• •••5 •••3 • • •2 ••••••. 9D. )D _D !WD !!D !@D. !#D. •3. 7611 • • • •8. 19743 ••••5 3 • 4 • •7 •. ••••••. -D. 2 416 48 0. 7548 54. 19 .. 2... .. .. 76o9eo;o|bj’da[ !WF zqoIa[gruj,0Bo !#& 9kd9eo;ooAoD 4k,;jk9eo;ooAoc,jo gmqjk.fL 9qj’-uhc9’;jk @ $&( × $W( = 2849 × #&! X 3fp[+.sh7yfw]jzqo76o0v’r;d,ao ?D mhk; [q;7e ,uglNvlk,zno c]t 3lh’luj38D 4k,;jk]k;,u9admk’g]nvdgrnjvo5j’g7njv’ faj’djk;3fp[+.sh-EdaoL l;o8vo|bj’gxaoI6[lujc9lkdF ,ugoNvmju &&#& m2 c]t 0hk’d;hk’cmdwfh $& mD 4k,;jk];’pk;0v’l;o8vooAo,u9adc,afL ok’ [5o,u 9af\kddhP’ #*W |j;p.lj !@ cdafF 3fp.shc8j]tcdaf[ao959eo;o \kddhP’gmqjkdaoD 4k,;jkc8j]tcdaf,u\kddhP’9ad|j;pL dq[8q;|bj’g8Aolt\jeltg\u !@ gmnjv 8k,mk’-njF 9kd95f Ο g4y’95f Α muj16jsjk’ )$W cm 9kd Ο D 4k,;jkdq[g8Aogmnjv|bj’wxwfhwd9ad-a’8uc,afL xB,0Po _ sq; ]k7k !#&WW du[ c]t xB,0Po-toyffP;dao * sq; ]k7k _ WWW du[D 9qj’7yfw]jj]k7kxB,0Po-toyffP;daooU !& sq;D g0Qk[kg] !WW dy3]dik, zt]yfcxh’wfh (& dy3]dik,D cxh’g0Qk[kg] @& dy3]dik, zt]yfg0Qk9ujwfh #W dy3]dik,D 4k,;jkgrnjvzt]yfg0Qk9uj $&W dy3]dik,9t8hv’.-hg0Qk [kg]9addy3]dik,L ]qf9ad7ao|bj’]k7k @@ (&W WWW du[F xtdkf0kpzjvofaj’oUG 9jkp,N,kIa[]qf $ WWW WWW du[F lj;opa’gs^nv9jkpzjvoc8j]tgfnvogmqjkdao rkp.o|bj’xuD 4k,;jk c8j]tgfnvo9jkp8nj,gmqjk.fL. ສດິ. *D (D. ×. •357 7 •8 34 • 56 •••4 ••••9 ••••••• ×. ••••. lD 653 2 173 2.2 58. &D. • • •1 ••• • • • •8. ×. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. $D. 37.

(42) [qfmu * dko76o c]t dkosko9eo;o4h;o !D dyf9tde dyf9tdemu ! 9eo;o4h;o[;d c]t 9eo;ome,t-kfc,jo9eo;ofP;daoF gryjo7yfw]jzqo76olv’ 9eo;o4h;o[;d7nda[7yfw]jzqo76olv’9eo;ome,t-kffaj’oUG 5 × 3 = 5 + 5 + 5 =15 D dD 9qj’xtdv[[A’l6f[A’ # ]5j,oUD –#. 3 × 5 = 15 3 × 4 = 12 3× 3 = 9 3× 2 =. ສດິ. –#. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 3 ×1 = 3× 0 = 3 × ( −1) = 3 × (−2) = 3 × (−3) =. 0D gIqk7yfw]jwfhvudco;|bj’faj’oUG X−#? × # = X–#? + X–#? + X–#? = – _ 9qj’7yfw]jzqo76o]5j,oUG a. X−#? × $ = DDDDDDDDDD S b. X−&? × $ = DDDDDDDDDDDD S. c. 6 × X–$? = d.. DDDDDDDDDDD S ( × X–5? = DDDDDDDDDDDD. 7D gIqkla’gdfgsaozqo76o0v’lv’9eo;o4h;omuj,ug7njv’\kp8jk’daogxao7nco;.fL. 38.

(43) dyf9tdemu @. dD 9qj’xtdv[[A’l6f[A’ X−#? ]5j,oUD (−3) × 4 = −12 (−3) × 3 = − 9 (−3) × 2 = − 6. +# +#. (−3) × 1 = (−3) × 0 = (−3) × (−1) = (−3) × (−2) =. 0D gIqkla’gdfgsaozqo76o0v’lv’9eo;o4h;o]q[gxao7nco;.fL 7D 9kdzqowfhIa[.odyf9tde ! c]t @ 9qj’xtdv[xt3spd]5j,oUD zqo76o0v’lv’9eo;o4h;omuj,ug7njv’\kpfP;daogxao DDDDDDDDDDDDDDDDD zqo76o0v’lv’9eo;o4h;omuj,ug7njv’\kp8jk’daogxao DDDDDDDDDDDDDDDDD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. dyf9tdemu # !D gIqkI6hc]h;;jkG !& = & × # 4vowfh !& ÷ & = # s^n !& ÷ # = & 9qj’8nj,9eo;omuj4nd8hv’.lj[jvo9Eg,afD dD !@ = # × DDDDDDDD 4vowfh !@ ÷ # = DDDDDDDDDDDD s^n !@ ÷ DDDDDDDD = DDDDDDDDDD 4vowfh @& ÷ & = DDDDDDDDDDDD 0D @& = & × DDDDDDDD 7D !@ = X−#? × DDDDDDDD 4vowfh !@ ÷ X− #? = DDDDDDDDDDDD s^n !@ ÷ DDDDDDDD = DDDDDDDDDD ’D @W = X−&? × DDDDDDDD 4vowfh @W ÷ X− &? = DDDDDDDDDDDD s^n @W ÷ DDDDDDDD = DDDDDDDDDD s^n X−!@? ÷ DDDDDDDD = DDDDDDDDDD 9D X−!@? = $ × DDDDDDDD 4vowfh X−!@? ÷ $ = DDDDDDDDDDDD lD X−!)? = _ × DDDDDDDD 4vowfh X−!)? ÷ _ = DDDDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDDD = DDDDDDDDDD -D X−#*? = X−$? × DDDDDDDD 4vowfh DDDDDDDDD ÷ X− $? = DDDDDDDDD s^n DDDDDDDDDDD ÷ DDDDDDDD = DDDDDDDD pD X−@$? = X−)? × DDDDDDDD 4vowfh DDDDDDDDDD ÷ X− )? = DDDDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDDD = DDDDDDDDDD fD X−@(? = −# × DDDDDDDD 4vowfh DDDDDDDDDD ÷ DDDDDDDDDDD = DDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDD = DDDDDDDDDD @D 9qj’xtdv[xt3spd ]5j,oUD zqoskolv’9eo;o4h;omuj,ug7njv’\kpfP;daogxao DDDDDDDDDDDD zqoskolv’9eo;o4h;omuj,ug7njv’\kp8jk’daogxao DDDDDDDDDDDD. 39.

(44) @D .97;k, !D dko76o9eo;o4h;o sâddkoG grnjv7yfw]jzqo76o0v’lv’9eo;o4h;oF gryjo76ow]ptsjk’9kd W 0v’r;d ,aoc]h;deoqfg7njv’\kp0v’zqo76ofaj’oUG zqo76olv’9eo;o4h;omuj,ug7njv’\kpfP;daogxao9eo;o[;dD zqo76olv’9eo;o4h;omuj,ug7njv’\kp8jk’daogxao9eo;o]q[D b. (−8) × (−3) = 24 8q;1jk’G a. 7 × 5 = 35; c. 9 × (−3) = − 27;. d . (−4) × 6 = − 24. zqogonjv’D zqo76o9eo;o4h;o|bj’.sh8q;,aogv’gvUo;jk < de]a’lv’< 0v’9eo;ooAoD .sh9eo;o4h;o a, zqo76o a × a 0Posp=h a × a = a2 vjko;jk < a de]a’lv’ < 8q;1jk’G dD )@= ) × ) = *$S 0D X−&?@ = X−&? × X−&? = @&D. ສດິ. a2 ≥ 0 D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. m5dM 9eo;o4h;o a gIqkwfh. @D 75o]adltot dko76o9eo;o4h;o,u]adltotla[xjPo[jvowfh c]t 3I,\6jwfhD .sh9eo;o4h;o8jk’l6o8k,.9 a; b; c gIqkwfhG a × b = b × a. a × b × c = a × ( b × c) = (a × b ) × c. 8q;1jk’G dD !& × @ = @ × !& = #WS 0D X− )? × $ = $ × X− )?= − #@D 7D & × X− #? × * = [& × X− #? ] × * = & × [X− #? × *] = X− !&? × * = & × X− !)? = − _WD ’D & × $ × ( = X& × $? × ( = & × X$ × (? = @W × ( = & × @) = !$W 9D X−!W? × X− #? × X−$? = [X−!W? × X− #? ] × X−$? = X−!W? × [X− #? × X−$?] = #W × X−$? = X−!W? × !@ = − !@WD 40.

(45) #D dko76o0v’s^kp9eo;o4h;o zqo76o0v’s^kp9eo;o4h;o8jk’l6ogxao9eo;o[;dF 4hkskd;jk9eo;olj;o76o]q[ gxao9eo;o76jD zqo76o0v’s^kp9eo;o4h;o8jk’l6ogxao9eo;o]q[F 4hkskd;jk9eo;olj;o76o]q[ gxao9eo;o7udD grnjv7yfw]jzqo76o0v’s^kp9eo;o4h;o8jk’l6oF 7;odeoqfg7njv’\kpdjvoc]h;7yfw]j zqo76o w]ptsjk’9kd W 0v’m5dlj;o76oD 8q;1jk’G A = (−20) × 3 × (−2) × (−4) × 5 S ,ulk,lj;o76o]q[7nG − @WS − @S − $F = − (20 × 3 × 2 × 4 × 5) g-yj’ 9eo;olj;o76o]q[gxao9eo;o7ud faj’oAo = − 2 400 zqo76ogxao9eo;o]q[D B = 3 × (−2) × (−5) × (−4) × (−1) × 6. ສດິ. = 3 × 2 × 5 × 4 × 1× 6. S ,ulujlj;o76o]q[7nG − @S − &S −$S −! g-yj’ 9eo;olj;o76o]q[gxao9eo;o76j faj’oAo zqo76ogxao9eo;o[;dD. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. = 720. $D dkosko9eo;o4h;o sâddkoG grnjv7yfw]jzqosko0v’lv’9eo;o4h;oF gryjoskow]ptsjk’9kd W 0v’ r;d,ao c]h;deoqfg7njv’\kp0v’zqoskofaj’oUG zqosko0v’lv’9eo;o4h;omuj,ug7njv’\kpvaofP;dao gxao9eo;o[;dD zqosko0v’lv’9eo;o4h;omuj,ug7njv’\kp8jk’dao gxao9eo;o]q[D \kp7;k,;jkG m5dM9eo;o4h;o8k,.9 a c]t b 3fp;jk b ≠ 0 S 4hk a c]t b ,ug7njv’\kpfP;dao (a ÷ b) ≥ 0 4hk a c]t b ,ug7njv’\kp8jk’dao (a ÷ b) ≤ 0 . 8q;1jk’G. dD @) ÷ ( = $S 0D X− @(? ÷ X− _? = #S. 7D X− @$? ÷ # = − )S ’D !@ ÷ X− $? = − #D. 41.

(46) &D dkoskogvu d]uf (Euclide) dkosko gvu d]ufc,jodkoskolv’9eo;ome,t-kf g-yj’zqosko c]t 8q;glf gxao9eo;ome,t-kf s^n l6oD .sh9eo;ome,t-kf a c]t b, a > b gIqkwfh a = b×q + r 3fp;jk r < b. 8q;1jk’G 342 16 gIqkwfh 342 = 16 × 21+ 6 22 21 6. *D dkosko0kfD 4hk8q;glf.odkosko gvu d]uf a .sh b skdgmqjkl6oD gryjog;Qk;jk a sko0kf.sh b c]t gIqkwfh a = b×qD 8q;1jk’G #* $ gryjog;Qk;jk 36 sko0kf.sh 4 c]t gIqkwfh 36 = 4×9D. a. c,jomt;u76o0v’ b. mP[gmqjk. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. \kpgsfG 4hk a sko0kf.sh b gryjog;Qk;jk < a c,jomt;u76o0v’ b < c]t < b c,jov5xt76o0v’ a < faj’oAogIqklts^5[wfhdkorq;raofaj’oUG. ສດິ. 0 9. b c,jov5xt76o0v’ a. ,ao,u9eo;ome,t-kf8jk’l6o k 3fp;jk a = b × kD 9eo;ome,t-kf|bj’sko0kf.sh @ wfhd=8+g,njv8q;g]dsq;|j;p0v’,aoc,joG WS @S $S *S )D 9eo;ome,t-kf|bj’sko0kf.sh & wfhd=8+g,njv8q;g]dsq;|j;p0v’,aoc,joG W c]t &D 9eo;ome,t-kf|bj’sko0kf.sh # wfhd=8+g,njvzqo[;d8q;]dma’\qfmuj.-h0Po9e o;ooAosko0kf.sh #D 9eo;ome,t-kf|bj’sko0kf.sh _ wfhd=8+g,njvzqo[;d8q;g]dma’\qfmuj.-h0Po9e o;ooAosko0kf.sh _D 9eo;ome,t-kf|bj’sko0kf.sh $ wfhd=8+g,njv9eo;omujxtdv[fh;plv’8q;g]d l5fmhkpsko0kf.sh $D 42.

(47) 8q;1jk’G dD !#!* sko0kf.sh $ grkt !* sko0kf.sh $D 0D !#!$ sko.sh $ [+0kf grkt !$ sko.sh $ [+0kfD sâdgdoG 9eo;ome,t-kf|bj’sko0kf.sh a c]t b 8hv’sko0kf.shzqo76o0v’r;d,aoD 8q;1jk’G ()* sko0kf.sh @ grkt;jk8q;g]dsq;|j;pc,jo *D ()* sko0kf.sh # grkt;jk ( + ) + * = @! c]t @! sko0kf.sh # phvo;jk @ + ! = #D ltoAo ()* sko0kf.sh * grkt @ × # = *D. #D [qfg/ydsafD. ×. −*. −#. !). −$. −!. @. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. !D 9qj’7yfw]jzqo76o]5j,oUG dD & × X−*?S 9D X−!@? × _S fD X−*? × X−@&?S 0D X−(? × X−_?S lD ( × !!S 8D (× X−&? × X−)? × $ × X−!W?S 7D X−#? × #S -D X−!)? × X−*?S 4D X−$?@ × X−#?S ’D W × X−(?S pD ) × X−@&?S mD X−@? × X−*? × @W × X−&? × X−_?S @D 9qj’xtdv[8k8t]k’dko76o]5j,oU.sh4nd8hv’D &. −@. # $ #D 9qj’xtdv[8k8t]k’]5j,oU.sh4nd8hv’D X. & −_ −$ #. Y −@. * −# $. Z −!#. A=X×Y −!W. −!$ −&. ) 43. B=Y×Z. C=A×Z. D=X×B.

(48) $D .sh. −3. 2. vu’8k,dko7yfw]j0hk’oUF 9qj’xtdv[I6[]5j,oU.sh4nd8hv’D. −6. dD. 1. ‐1. 2. 0D. ‐2. 5. 7D. ‐6. ‐3. &D 9qj’7yfw]jzqo76o]5j,oU. 5. [. ‐2. D = (−4) 2 × (−3) × 2 2 × 6. A = (−8) × (−2) × (−5) × 4 B = [(−25) × 4 × (−2)]× 5 C = 5 × (−3) × [(−2) × (−10) × (−4)]. [. ]. ‐1. ]. E = 5 × 8 × (−1) × (−2) × (−4) 2 2. 2. F = (−2) × 3 × [4 × (−11)] 2. 2. (D .sh. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. *D 9qj’xtdv[ltg\uzqo]5j,oU.sh4nd8hv’D 9D X−$? V DDDDDDDD = WS dD # V DDDDDDDDD = − *S 0D X−@? V DDDDDDDDDD = −#WS lD $ V DDDDDD V X−)? = *$S 7D DDDDDDDDDD V X−#? = @(S -D X−$? V X−#? V DDDDDDD = (@S pD X−!#? V #& V DDDDDD V !! V X−(? = WS ’D X−&? V DDDDDDDDDDDD = − @&S vu’8k,dko7yfw]j0hk’oUF 9qj’xtdv[I6[]5j,oU.sh4nd8hv’D. a×b a. b −80. 50 15. 8 −2. −3. 4. )D 9qj’7yfw]jzqosko]5j,oUD dD X−!W? ÷ @S 0D @$ ÷ X−)?S 7D X−!*? ÷ X−)?S ’D # ÷ X−#?S 9D X−@(? ÷ X−_?S. lD !@ ÷ X−!?S -D #* ÷ !@S pD X−$&? ÷ X−&?S fD W ÷ X−(?S 8D W ÷ _D 44.

(49) _D 9qj’xtdv[ltg\uzqo]5j,oU.sh4nd8hv’D dD DDDDDDDD ÷ # = −_S 9D DDDDDDD ÷ X−$? = −#S lD DDDDDDDDDD ÷ X−#?= &S 0D X−@@? ÷ DDDDDDDD = !!S 7D X−)? ÷ DDDDDDD= $S -D DDDDDDDD ÷ $ = −&S ’D X−!@? ÷ DDDDDDD =−@S pD −#* ÷ DDDDDD = _S. −@. _. *W −!@. #@ −). 9D x ÷ (−6) = 6; lD @W ÷ x = −5; -D x ÷ (−8) = −3; pD −#* ÷ x = 3. !@D 9qj’7yfw]jzqosko]5j,oUD dD !&WW ÷ !WS 0D # $W@ WWW ÷ ! WWWS 7D !W WWW ÷ ! WWWS ’D @WW WWW ÷ !W@ S. 9D #W$WW ÷ X−!WW?S lD X−#@WW? ÷ X−!W?S -D X−&WWW? ÷ X−!W?@S pD X−#$ WWW? ÷ !WW. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. !!D 9qj’-vd7jk0v’ x le]a[ltg\uzqo]5j,oUD dD X−@*? ÷ x = !#S 0D x ÷ 8 = −3; 7D !$ ÷ x = @S ’D X−$_? ÷ X−(? = x S. ສດິ. !WD 9qj’xtdv[8k8t]k’]5j,oU.sh4nd8hv’D x − &* −!* $& y −( $ −!! x÷y −_ #. fD $_ ÷ DDDDDDDD = −(S 8D DDDDDDDD ÷ $ = −!@S oD X−&*? ÷ DDDDDDD = (S [D $@ ÷ DDDDDDDD = −*D. !#D 9qj’8A’[A’g]dc]h;7yfw]jzqoskoc[[gvud]yfD dD $@)$ ÷ *S ’D _* @@( ÷ (_S -D * *#W ÷ )&S 0D !@ &(@ ÷ !@S 9D *# W$# ÷ !_S pD !@ (*& ÷ **S 7D #&($ ÷ #&S lD (! W&W ÷ _@S fD $@ &)$ ÷ !*D !$D 3fp[+8hv’7yfw]jdkoskoF 9qj’[vd;jk9eo;o]5j,oU,u9eo;o.fsko.sh @S #S $S &S *S ) s^n _ 0kfD !$$S *WS !_#@S @@ &@*S $$#S &@&S !!(*S $&_S !) W!&S !@S @#S @$&WS !@ &$*S @$&S #$#D 45.

(50) [qfmu ( mt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f 0v’9eo;ome,t-kf !D dyf9tde dyf9tdemu ! dD 9qj’8nj,.lj8k8t]k’]5j,oU.sh4nd8hv’D ! @ # $ & * *. (. ). _. !W #W. …. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 9eo;o.oc4;mu @ wfh,kco;.fL 9eo;ofaj’djk;gvUo;jkco;.fL 0D 9qj’0Po9eo;ome,t-kfc8j ! g4y’ $@ c]h;0ufvhv,9eo;omujgxaomt;u76o0v’ $D 7D 9qj’0Po9eo;omujgxaomt;u76o0v’ $ sqd9eo;o8k,.9D ’D 9qj’7yfw]jzqo]q[lv’mt;u76o0v’ $ 4afdao8k,]efa[ 8k,.9,kluj76jF la’gdfzqowfhIa[gxao7nco;.f L. dyf9tdemu @ dD 9qj’0ufvhv,9eo;o]5j,oUmujgxaomt;u76o0v’ (D @!S @&S @)S @_S #&S #(S $@S $&S $)S *#S _!S _#S !$WS !$#D 0D 9qj’0Po9eo;omujgxaomt;u76o0v’ ( sqd9eo;o8k,.9 D 7D 4hk9eo;ome,t-kf a c,jomt;u76o0v’ (F gIqk0Po a gxaozqo76o0v’ ( faj’oAoG a = 7V DDDDD 4hk9eo;ome,t-kf a [+c,jomt;u76o0v’ (F gIqk0Po a gxaozqo76o0v’ ( faj’oAoG a = 7 V DDDDDDD + DDDDDDDDD 8q;glfgxao9eo;ome,t-kf|hvpd;jk (D d=]toumqj;wxG .sh9eo;ome,t-kf a c]t bF gIqkwfhG a c,jomt;u76o0v’ b 4hk;jk ,u9eo;ome,t-kf q g-yj’;jk a = bq. 4vowfh a ÷ b = q c]t 8q;glfgmqjk WD gryjog;Qk;jk a sko0kf.sh b 46.

(51) @D .97;k, dD mt;u76o0v’9eo;ome,t-kfD mt;u76o0v’9eo;ome,t-kf a c,jozqo76o0v’ a da[9eo;ome,t-kf !S @S #S DDD 8q;1jk’G d5j,mt;u76o0v’ # c,jo {#S *S _S !@S DDD} = 3 ` d5j,mt;u76o0v’ $ c,jo {$S )S !@S !*S DDD} = 4 ` 0D mt;u76oIj;,|hvpl5f 0v’ a c]t b 0Posp=h mI|l ( a, b ) s^n LCM ( a, b ) D gIqk,uG d5j,mt;u76o0v’ # c,jo {#S *S _S !@S !&S !)S @!S @$S @(S DDD}= 3 ` d5j,mt;u76o0v’ $ c,jo {$S )S !@S !*S @WS @$S @)S #@S #*S DDD}= 4 ` d5j,mt;u76oIj;,0v’ # c]t $ c,jo {!@S @$S #*S $)S *WS DDD}= 12 ` mt;u76oIj;,|hvpl5f0v’ # c]t $ c,jo !@ 0Posp=h LCM ( 3, 4 ) = 12 D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. gIqk,uG d5j,mt;u76o0v’ * c,jo {*S !@S !)S @$S #WS #*S $@S $)S DDD} = 6 ` d5j,mt;u76o0v’ ) c,jo {)S !*S @$S #@S $WS $)S &*S *$S DDD} = 8 ` d5j,mt;u76oIj;,0v’ * c]t ) c,jo {@$S $)S (@S _*S DDD}= 24 ` mt;u76oIj;,|hvpl5f0v’ * c]t ) c,jo @$ 0Posp=h mI|lX*F )? = @$ s^n LCM ( 6,8 ) = 24 D. 7D v5xt76o0v’9eo;ome,t-kf .shlv’9eo;ome,t-kf a c]t bF 4hk a sko0kf.sh b D gIqkg;Qk;jk b c,jo v5xt76o0v’ aD 8q;1jk’G #* sko0kf.sh $F gIqkwfh $ c,jov5xt76o0v’ #*D 9qj’-vd9eo;ome,t-kfma’\qfmujgxaov5xt76o0v’ #*D gIqk,uG #* = ! V #*S #* = @ V !)S #* = # V !@S #* = $ V _S #* = * V * faj’oAo v5xt76o0v’ #* ,u G !S @S #S $S *S _S !@S !)S #*D d5j,v5xt76o0v’ #* c,jo {!S @S #S $S *S _S !@S !)S #*}. 47.

(52) sâdgdoG m5dM9eo;omujgxaov5xt76o0v’9eo;o a c,jov5xt76o0v’m5dM9eo;omujgxao mt;u76o0v’ a. m5dMmt;u76o0v’ a sko0kf.shv5xt76om5dM8q;0v’ a. 8q;1jk’G { mt;u76o0v’ * c,jo *S !@S !)S @$S #WS DDD c]t v5xt76o0v’ * c,jo !S @S # c]t *. { mt;u76o0v’ * c,jo *S !@S !)S @$S #WS DDD ]h;oc8jsko0kf.sh !S @S # c]t * g-yj’ c,jov5xt76o0v’ * D. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. 75o]adltot m5dM9eo;ome,t-kf nF gIqkwfh n sko0kf.sh ! c]t 8q;,aogv’ ltoAo v5xt76o|hvpl5f0v’ n c,jo ! c]t v5xt76os^kpl5f0v’ n c,jo8q;,aogv’7n nD m5dM9eo;ome,t-kf nF 4hk b c,jov5xt76o0v’ n gIqkwfh 1 ≤ b≤ n D ltoAo 9eo;ov5xt76o0v’9eo;ome,t-kf n oa[wfh XlYol5f?D. dko-vdskv5xt76o0v’9eo;ome,t-kfgIqkxt8y[affaj’oUG 8q;1jk’G -vdskv5xt76o0v’ !@D !@ = ! V !@ = @ V * = # V $D gIqk15fdko7yfw]j 4hkgsao;jklj;o76oxUoda[7noD gIqkwfh v5xt76o0v’ !@ c,jo !S @S #S $S * c]t !@ g-ajofP;dao le]a[ !)D !) = ! V !) = @ V _ = # V *D gIqkwfh v5xt76o0v’ !) c,jo !S @S #S *S _ c]t !) ’D v5xt76oIj;,s^kpl5f 0v’ a c]t b 0Posp=h vIs^l ( a, b ) s^n GCD ( a, b ) D gIqkI6h d5j,v5xt76o0v’ !@ c,jo {!S @S #S $S *S !@ } c]t d5j,v5xt76o0v’ !) c,jo {!S @S #S *S _S !) } gIqkgsao;jkG v5xt76oIj;,0v’ !@ c]t !) ,uG !S @S # c]t *D le]a[ * c,jov5xt76oIj;,muj s^kpd;jk\6j0v’!@ c]t !)D gryjogvUo * ;jk v5xt76oIj;,s^kpl5f0v’!@ c]t !) 0Posp=h vIs^lX!@F !) ? = * s^n GCD (12,18) = 6 D 48.

(53) gIqkI6hG d5j,v5xt76o0v’ @$ c,jo {!S @S #S $S *S )S !@S @$}D d5j,v5xt76o0v’ #* c,jo {!S @S #S $S *S _S !@S !)S #*} c]t d5j,v5xt76o0v’ $) c,jo {!S @S #S $S *S )S !@S !*S @$S $)} d5j,v5xt76oIj;,0v’ @$S #* c]t $) c,jo {!S @S #S $S *S !@ }D ltoAo GCD ( 24,36, 48 ) = 12 D. ສດິ. sâdgdo ! .sh9eo;ome,t-kf a, b c]t cF 4hk;jk GCD ( a, b ) = t c]h; GCD ( a, b, c ) = GCD ( t , c ) . sâdgdo@ .sh9eo;ome,t-kf a, bF b < a c]t r c,jo8q;glf0v’dkosko gvu d]uf a .sh bD 4hk;jk r = W c]h; GCD ( a, b ) = b . 4hk;jk r ≠ W c]h; GCD ( a, b ) = GCD ( b, r ) .. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 9D dko-vdskv5xt76oIj;,s^kpl5f0v’lv’ s^n s^kp9eo;ome,t-kfD sâddkoG v5xt76oIj;,s^kpl5f0v’lv’9eo;ome,t-kf a, b c]t b < a, c,jo 8q;glf8jk’l6o 8q;l5fmhkp0v’dkoskogvud]yf8+gonjv’dao0v’ a .sh bD g-ajo;jkG a sko.sh b c]h;glf r ≠ WS gIqksko b .sh r c]h;glf r′ ≠ WS gIqksko r .sh r′ c]h;glf r″ ≠ WS gIqksko r′ .sh r″ c]h;glf r′′′ = W D gIqkwfhG GCD ( a, b ) = GCD ( b, r ) = GCD ( r , r ') = r " D 8q;1jk’G !D -vdsk GCD (12,18) D !) !@ !@ * gIqkwfh GCD (12,18) = 6. 6 1. 0 2. @D -vdsk GCD ( 24,32, 48) D gIqk-vd GCD ( 24,32 ) D gIqkwfhG #@ @$ c]t @$ ) F faj’oAo GCD ( 24,32 ) = 8. gIqk-vd ltoAo. GCD ( 8, 48 ). 0 3. 8 1. gIqkwfhG $) ) 0 6. faj’oAo. GCD ( 24,32, 48 ) = GCD ( 8, 48 ) = 8 D. 49. GCD ( 8, 48 ) = 8 D.

(54) ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. lD 9eo;o,6o 9eo;ome,t-kfmuj,uv5xt76orP’c8jlv’8q; 7n ! c]t 8q;,aogv’gmqjkoAo gvUo;jk 9eo;o,6oD 8q;1jk’G !# ,uv5xt76o ! c]t !# gmqjkoAoD ltoAo !# c,jo9eo;o,6oD & ,uv5xt76o ! c]t & gmqjkoAoD ltoAo & c,jo9eo;o,6oD $ ,uv5xt76o !F @ c]t $. ltoAo $ [+c,jo9eo;o,6oD !@ ,uv5xt76o !F @F #F $F * c]t !@. ltoAo !@ [+c,jo9eo;o,6oD 9eo;o,6omuj|hvpd;jk !WWD gryjo-vdsk9eo;o,6omuj|hvpd;jk !WW faj’oUG 0Po9eo;ome,t-kfc8j ! 1 2 3 4 5 6 7 8 9 10 sk !WW .lj8k8t]k’10 ×10 F 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0hk !F 0hkmt;u76o0v’ @F #F &F 31 32 33 34 35 36 37 38 39 40 ( c]t !!D 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9eo;omujgs^nv.o8k8t]k’ 61 62 63 64 65 66 67 68 69 70 c,jo9eo;o,6o c8j ! sk !WW 71 72 73 74 75 76 77 78 79 80 g-yj’c,jo @F #F &F (F !!F !#F 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 !(F !_F @#F @_F #!F #(F $!F $#F $(F &#F &_F *!F *(F (!F(#F (_F )#F )_ c]t _(D 75o]adltot m5d9eo;ome,t-kfmuj[+c,jo9eo;o,6o,u1jk’|hvpv5xt76ogxao9eo;o,6o|bj’8q;D grnjvd;fg[yj’9eo;ome,t-kf|bj’c,jo9eo;o,6o s^n [+F gryjosko9eo;ooAo.sh 9eo;o,6o]Po8yfdaoD 4hkdkoskoskd[+0kf c]t gryjo9t15fdkoskog,njvwfh zqoskogmqjkda[ s^n |hvpd;jk8q;sko c]t gIqkwfh;jk 9eo;ooAoc,jo9eo;o,6oD g;Qkvud1jk’|bj’G 1kdI6h;jk9eo;ome,t-kf A gxao9eo;o,6o s^n [+F gryjosko A .sh9eo;o,6o]Po4afdao 4hkskdsko[+0kfIvf9eo;o,6o P c]t P2 > A c,jo A gxao9eo;o,6oD 8q;1jk’G _( sko.sh @S #S & [+0kfD gIqksko.sh (S !!S !#D gIqkwfhG _( = ( V !# + * _( = !! V ) + _S _( = !# V ( + *D ltcf’;jk _( c,jo9eo;o,6oD 50.

(55) -D dko0Po9eo;ome,t-kfgxaozqo76o0v’9eo;o,6oD 9eo;ome,t-kfm5dM8q;lk,kf0Pogxaozqo76o0v’9eo;o,6owfhD c8j]trqf 0v’zqo76ofaj’djk; gvUo;jk lj;o76o,6oD 8q;1jk’G !& = # V &S @! = # V (S #!& = !& V @! = # V& V# V(= #@ V & V (S #* = $ V _ = @@ V #@D ;ymu-vdlj;o76o,6o0v’9eo;ome,t-kfF gryjosko9eo;ooAo.sh9eo;o,6o3fp g]uj,9kd9eo;o,6o|hvpl5fmujsko.sh9eo;ooAo0kf c]t skozqoskowfh,k8+wx g]NvpMD 105 3 35 5 7 7 1. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. 18 2 9 3 3 3 1. ສດິ. 8q;1jk’G dD #!& # gIqkwfhG #!& = #@ V & V (D 0D #* @ gIqkwfhG #* = @@ V #@D. dkooe.-hlj;o76o,6ogrnjv-vdsk @. v5xt76oIj;,s^kpl5f c]t mt;u76oIj;,|hvpl5fD 8q;1jk’G #!& = # V & V ( c]t !W) = @@ V ##. mt;u76oIj;,|hvpl5f c,jozqo76o0v’lj;o76o,6omuj8jk’daoma’\qf.shlj;o76o,6o mujIj;,daoD le]a[lj;o76o,6omujIj;,dao .shgvqklj;o76omuj,ude]a’.spjd;jk\6jD LCM ( 315,108 ) = 22 × 33 × 5 × 7 = 3780.. v5xt76oIj;,s^kpl5f c,jozqo76o0v’lj;o76o,6omujIj;,daoma’\qf0v’r;d,ao Xc8jgvqklj;o76o,6oIj;,daomuj,ude]a’|hvpd;jk\6j? GCD ( 315,108 ) = 32 = 9. pD lv’9eo;o,6o8+daoD lv’9eo;ome,t-kfmuj,uv5xt76oIj;,s^kpl5fc,jo ! gvUo;jk lv’9eo;o ,6o8+daoD \kp7;k,;jk a c]t b gxao9eo;o,6o8+dao g,njv GCD ( a, b ) = 1. 8q;1jk’G GCD (8,13) = 1 F ltoAo 8 c]t 13 gxao9eo;o,6o8+daoD 51.

(56) #D [qfg/ydsaf. ສ. ວ ສະ .ສ ຫງ ວນ ລຂິ ະ. ສດິ. !D 9qj’d;fg[yj’9eo;o]5j,oU;jk9eo;o.fgxao9eo;o,6oD $#S $(S $_S *#S *(S *_S (&S (_S )(S )_S _(S _)S !!#S !!(S @!(S #((S !#(_S $#(#D @D dD 9qj’[vd9eo;o,6omuj|hvpd;jk #WD 0D 9qj’0Po9eo;o]5j,oUgxaozqo76o0v’9eo;o,6oD )$S !@!S @&WS @_$S !)WS @$_S )*$S @&@WS (_@WS )WWWS &($WS #* V $@S (@ V((S !W) V(&S )$ V@& V!@!S #* V @( V!$@S *& V $_ V@$D #D dD 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ *S ) c]t !WD 0D 9qj’[vdmt;u76oIj;,!W 8q;mevyf0v’ * c]t ) rhv,ma’[vdmt;u76oIj;,|hvp l5f0v’r;d,aoD 7D 9qj’[vdmt;u76oIj;,!W 8q;mevyf0v’ * c]t !W rhv,ma’[vdmt;u76oIj;,|hvp l5f0v’r;d,aoD ’D 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ !W c]t ) rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD 9D 9qj’[vdmt;u76oIj;, & 8q;mevyf0v’ *S ) c]t !W rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD $D dD 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ &S ( c]t !&D 0D9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ & c]t !& rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD 7D 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ & c]t ( rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD ’D 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ !& c]t ( rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD 9D 9qj’[vdmt;u76oIj;, & 8q;mevyf0v’ &S ( c]t !& rhv,ma’[vdmt;u76oIj;, |hvpl5f0v’r;d,aoD &D 9qj’-vdskG LCM ( 36, 42 ) F LCM (120,96 ) F LCM (18, 22 ) F LCM ( 82,52 ) F LCM ( 250,150 ) F LCM ( 254, 672 ) F LCM ( 720,900 ) F LCM (14, 26,32 ) F LCM (15, 75,36 ) c]t LCM ( 84, 62,56, 42 ) D 52.

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