{\rtf1\ansi\ansicpg1250\uc1 \deff0\deflang1033\deflangfe1033{\fonttbl{\f0\froman\fcharset0\fprq2{\*\panose 02020603050405020304}Times New Roman;}{\f1\fswiss\fcharset0\fprq2{\*\panose 020b0604020202020204}Arial;}
{\f2\fmodern\fcharset0\fprq1{\*\panose 02070309020205020404}Courier New;}{\f3\froman\fcharset2\fprq2{\*\panose 05050102010706020507}Symbol;}{\f14\fnil\fcharset2\fprq2{\*\panose 05000000000000000000}Wingdings;}
{\f19\fswiss\fcharset0\fprq2{\*\panose 020b0807020203060204}BankGothic Md BT{\*\falt Univers Condensed};}{\f70\froman\fcharset238\fprq2 Times New Roman CE;}{\f71\froman\fcharset204\fprq2 Times New Roman Cyr;}
{\f73\froman\fcharset161\fprq2 Times New Roman Greek;}{\f74\froman\fcharset162\fprq2 Times New Roman Tur;}{\f75\froman\fcharset186\fprq2 Times New Roman Baltic;}{\f76\fswiss\fcharset238\fprq2 Arial CE;}{\f77\fswiss\fcharset204\fprq2 Arial Cyr;}
{\f79\fswiss\fcharset161\fprq2 Arial Greek;}{\f80\fswiss\fcharset162\fprq2 Arial Tur;}{\f81\fswiss\fcharset186\fprq2 Arial Baltic;}{\f82\fmodern\fcharset238\fprq1 Courier New CE;}{\f83\fmodern\fcharset204\fprq1 Courier New Cyr;}
{\f85\fmodern\fcharset161\fprq1 Courier New Greek;}{\f86\fmodern\fcharset162\fprq1 Courier New Tur;}{\f87\fmodern\fcharset186\fprq1 Courier New Baltic;}{\f188\fswiss\fcharset162\fprq2 BankGothic Md BT Tur{\*\falt Univers Condensed};}}
{\colortbl;\red0\green0\blue0;\red0\green0\blue255;\red0\green255\blue255;\red0\green255\blue0;\red255\green0\blue255;\red255\green0\blue0;\red255\green255\blue0;\red255\green255\blue255;\red0\green0\blue128;\red0\green128\blue128;\red0\green128\blue0;
\red128\green0\blue128;\red128\green0\blue0;\red128\green128\blue0;\red128\green128\blue128;\red192\green192\blue192;}{\stylesheet{\nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid \snext0 Normal;}{
\s1\sb240\sa60\keepn\nowidctlpar\widctlpar\adjustright \b\f1\fs28\lang1048\kerning28\cgrid \sbasedon0 \snext0 heading 1;}{\*\cs10 \additive Default Paragraph Font;}{\s15\nowidctlpar\widctlpar\tqc\tx4153\tqr\tx8306\adjustright \fs20\lang1048\cgrid 
\sbasedon0 \snext15 header;}{\s16\nowidctlpar\widctlpar\tqc\tx4153\tqr\tx8306\adjustright \fs20\lang1048\cgrid \sbasedon0 \snext16 footer;}{\*\cs17 \additive \sbasedon10 page number;}{\s18\nowidctlpar\widctlpar\adjustright \f2\fs20\lang1048\cgrid 
\sbasedon0 \snext18 Plain Text;}}{\*\listtable{\list\listtemplateid201916417\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f3\fbias0 \fi-360\li360\jclisttab\tx360 }
{\listname ;}\listid272053824}{\list\listtemplateid201916427\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3880 ?;}{\levelnumbers;}\f14\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname 
;}\listid369494833}{\list\listtemplateid201916417\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f3\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname 
;}\listid401950534}{\list\listtemplateid201916417\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f3\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname 
;}\listid410851262}{\list\listtemplateid201916421\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3929 ?;}{\levelnumbers;}\f14\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname 
;}\listid447818851}{\list\listtemplateid201916421\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3929 ?;}{\levelnumbers;}\f14\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname 
;}\listid457257315}{\list\listtemplateid201916431\listsimple{\listlevel\levelnfc0\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'02\'00.;}{\levelnumbers\'01;}\fi-360\li360\jclisttab\tx360 }{\listname ;}\listid870268268}
{\list\listtemplateid201916417\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f3\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname ;}\listid882131664}
{\list\listtemplateid201916417\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f3\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname ;}\listid1778209477}
{\list\listtemplateid201916421\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\'01\u-3929 ?;}{\levelnumbers;}\f14\fbias0 \fi-360\li360\jclisttab\tx360 }{\listname ;}\listid1889106486}}
{\*\listoverridetable{\listoverride\listid870268268\listoverridecount0\ls1}{\listoverride\listid410851262\listoverridecount0\ls2}{\listoverride\listid401950534\listoverridecount0\ls3}{\listoverride\listid272053824\listoverridecount0\ls4}
{\listoverride\listid1778209477\listoverridecount0\ls5}{\listoverride\listid369494833\listoverridecount0\ls6}{\listoverride\listid447818851\listoverridecount0\ls7}{\listoverride\listid457257315\listoverridecount0\ls8}{\listoverride\listid1889106486
\listoverridecount0\ls9}{\listoverride\listid882131664\listoverridecount0\ls10}}{\info{\title Rezolvarea ecua\'feiilor diofantice}{\author Raluca Musaloiu-Elefteri}{\operator Raluca Musaloiu-Elefteri}{\creatim\yr1999\mo7\dy8\hr17\min25}
{\revtim\yr1999\mo7\dy8\hr17\min25}{\printim\yr1999\mo1\dy13\hr1\min7}{\version2}{\edmins0}{\nofpages5}{\nofwords847}{\nofchars4832}{\nofcharsws5934}{\vern71}}\paperw11906\paperh16838\margt1135 
\widowctrl\ftnbj\aenddoc\formshade\viewkind1\viewscale100\pgbrdrhead\pgbrdrfoot \fet0\sectd \linex0\headery709\footery709\colsx709\endnhere\sectdefaultcl {\footer \pard\plain \s16\qc\nowidctlpar\widctlpar\brdrt\brdrs\brdrw10\brsp20 
\tqc\tx4153\tqr\tx8306\adjustright \fs20\lang1048\cgrid {- }{\field{\*\fldinst {\cs17  PAGE }}{\fldrslt {\cs17\lang1024 1}}}{\cs17 /}{\field{\*\fldinst {\cs17  NUMPAGES }}{\fldrslt {\cs17\lang1024 1}}}{\cs17  -}{
\par }}{\*\pnseclvl1\pnucrm\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl2\pnucltr\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl3\pndec\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl4\pnlcltr\pnstart1\pnindent720\pnhang{\pntxta )}}
{\*\pnseclvl5\pndec\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl8
\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \s1\qj\sb240\sa60\keepn\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f1\fs28\lang1048\kerning28\cgrid {
\f70 Rezolvarea ecua\'feiilor diofantice
\par }\pard\plain \s15\qj\nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid {
\par }{\f70 Orice congruen\'fe\'e3 }{\i ax}{\i\sub 1}{\i +c}{\i {\field{\*\fldinst SYMBOL 186 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i 0 (mod b)}{\f70  se poate scrie ca o ecua\'feie }{\i ax}{\i\sub 1}{\i +bx}{\i\sub 2}{\i +c=0}{ (\'een care }{\i a}{\i 
{\field{\*\fldinst SYMBOL 185 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i  0}{), }{\i b}{\i {\field{\*\fldinst SYMBOL 179 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i\f70 1 \'bai c, x}{\i\sub 1}{\i , x}{\i\sub 2}{\f70  sunt numere \'eentregi. Dac\'e3 }{\i 
a, b, c}{\f70  sunt numere \'eentregi date \'bai }{\i x}{\i\sub 1}{\i\f70  \'bai x}{\i\sub 2}{ sunt considerate necunoscute, problema se red}{\f70 uce la g\'e3sirea solu\'feiilor \'eentregi ale unei ecua\'feii liniare cu coeficien\'fei \'eentregi. Daca }{
\i f(x}{\i\sub 1}{\i ,\'85, x}{\i\sub n}{\i )}{ este un polinom \'een }{\i x}{\i\sub 1}{\i ,\'85, x}{\i\sub n}{\f70  cu coeficien\'fei \'eentregi, atunci ecua\'feia }{\i f(x}{\i\sub 1}{\i ,\'85, x}{\i\sub n}{\i ) = A}{\f70  se nume\'bate }{\b\f70 
diofantic\'e3 }{\f70 dac\'e3 solu\'feiile ei sunt numere \'eentregi. Denumirea acestor ecua\'feii deriv\'e3 de la numele matematicianului grec }{\b Diofantos}{\f70  din Alexandria. Dac\'e3 o astfel de ecua\'feie admite solu\'fe
ii, atunci ea admite o infinitate de }{\i n}{-upluri care o satisfac.
\par 
\par \'cen continuare se va trata cazul  n=2: }{\fs24 ax+by=c
\par }{
\par }{\f70 Dac\'e3 }{\i a}{\f70  \'bai }{\i b}{ sunt numere pr}{\f70 ime \'eentre ele \'bai }{\i x}{\i\sub 0}{, }{\i y}{\i\sub 0}{\f70  constituie o solu\'feie pentru }{\i ax+by=c}{\f70 , atunci totalitatea solu\'feiilor se poate reprezenta sub forma: }{\i 
x= x}{\i\sub 0}{\i +bt}{, }{\i y= y}{\i\sub 0}{\i  \endash at}{, unde }{\i t}{\f70  este un num\'e3r \'eentreg oarecare. O solu\'feie a ecua\'feiei se poate ob\'feine cu ajutorul penultimei frac\'feii de aproximare pentru reprezentarea sub form\'e3
 de frac\'feie continu\'e3 a lui }{\i a/b}{\f70 . Consider\'e2nd c\'e3 penultima frac\'feie este }{\i m/n}{, }{\i x}{\i\sub 0}{\i =nc, y}{\i\sub 0}{\i =-mc.
\par }{
\par }{\b\i Exemplu:
\par 
\par }{\f70 Fie ecua\'feia: 43x+19y=2.
\par }{
\par }{\f70 Frac\'feiile de aproximare ale lui 43/19 sunt: 7/3, 9/4, 43/19.
\par }{
\par }{\lang1024 {\shp{\*\shpinst\shpleft792\shptop63\shpright6732\shpbottom1282\shpfhdr0\shpbxcolumn\shpbypara\shpwr1\shpwrk0\shpfblwtxt0\shpz0\shplid1027{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pib}{\sv 
{\pict\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw10478\pich2150\picwgoal5940\pichgoal1219\wmetafile8\bliptag504379189\blipupi414{\*\blipuid 1e103735c7063503fa0b65fad26c48cd}
010009000003930200000400150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000
050000000b0200000000050000000c02a00720251200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0240000560700000b00000026060f00
0c004d617468547970650000a00209000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002dc0105000000
14020002c2050500000013020002400709000000fa02000008000000000000002200040000002d010100050000001402e803460b050000001302e803c40c0400
00002d0100000500000014020002260b0500000013020002e40c040000002d010100050000001402e8030813050000001302e803f013040000002d0100000500
000014020002ca1005000000130200021014040000002d010100050000001402d005541a050000001302d0053c1b050000001402e803341a050000001302e803
5c1b040000002d0100000500000014020002f61705000000130200027c1b040000002d010100050000001402d005ae23050000001302d0059624050000001402
e803a021050000001302e803b624040000002d0100000500000014020002621f0500000013020002d62415000000fb0280fe0000000000009001000000000402
001054696d6573204e657720526f6d616e000083040000002d01020008000000320a5a07c82301000000340008000000320a4005c22301000000310008000000
320a3006902101000000310008000000320a5803cb2201000000310008000000320a4804701f01000000330008000000320a7001bc2101000000310008000000
320a6002261d01000000320008000000320a5a076e1a01000000340008000000320a40056b1a01000000350008000000320a5803681a01000000310008000000
320a4804041801000000330008000000320a7001591901000000310008000000320a6002ba1501000000320008000000320a72051f1301000000350008000000
320a5803221301000000340008000000320a4804d81001000000330008000000320a70010d1201000000310008000000320a60028e0e01000000320008000000
320a7205a80b01000000350008000000320a5803360b02000000313908000000320a7001a50b01000000310008000000320a6002ea0801000000320008000000
320a8a03b20502000000313908000000320a7001240601000000350008000000320a6002860301000000320008000000320a8a033f0002000000313908000000
320a70015a0002000000343310000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01030004000000f001020008000000
320a30067e22010000002b0008000000320a48047020010000002b0008000000320a6002321e010000002b0008000000320a6002ee1b010000003d0008000000
320a48040419010000002b0008000000320a6002c616010000002b0008000000320a60028214010000003d0008000000320a4804d811010000002b0008000000
320a60029a0f010000002b0008000000320a6002560d010000003d0008000000320a6002f609010000002b0008000000320a6002b207010000003d0008000000
320a60029204010000002b0008000000320a60024e02010000003d000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000ee0102022253797374656d00ee040000002d01020004000000f0010300030000000000}
}}{\sp{\sn pictureId}{\sv 65536}}{\sp{\sn pictureActive}{\sv 0}}{\sp{\sn fLine}{\sv 0}}{\shptxt \pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid {\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid {\object\objemb
\objw5941\objh1219{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e33000000000000000000000e0000
d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffdffffff04000000feffffff05000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c00000000000004600000000000000000000000080e8
27b14dc9be0103000000800200000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
0000000000000000000000001400000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002010100000003000000ffffffff0000000000000000000000000000000000000000000000000000
0000000000000000000001000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff04000000ffffffff000000000000000000000000000000000000000000000000
000000000000000000000000030000000400000000000000feffffff02000000fefffffffeffffff0500000006000000070000000800000009000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c0000000000000461700
00004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000001c000000020032d72c01000000000000d06b4900c47649000000000003010103000a01030e000001028834000288330000010288310002883900000002863d000288320002862b00030e00000102883500000102883100028839
00000002863d000288320002862b00030e000001028831000001030e00000102883100028839000001028835000000000002863d000288320002862b00030e0000010288310000010288330002862b00030e000001028834000001028835000000000002863d000288320002862b00030e00000102883100000102883300
02862b00030e000001028831000001030e0000010288350000010288340000000000000002863d004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffffffffffffffff000000000000
0000000000000000000000000000000000000000000000000000000000000400000048010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff00000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff0000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000288320002862b00030e0000010288310000010288330002862b00030e0000010288310000010288310002862b00030e000001028831000001028834000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001050000050000000d0000004d45544146494c455049435400ee2800009af7ffff300500000800ee2866080000
010009000003930200000400150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02a00720251200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0240000560700000b0000002606
0f000c004d617468547970650000a00209000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002dc010500000014020002c2050500000013020002400709000000fa02000008000000000000002200040000002d010100050000001402e803460b050000001302e803
c40c040000002d0100000500000014020002260b0500000013020002e40c040000002d010100050000001402e8030813050000001302e803f013040000002d0100000500000014020002ca1005000000130200021014040000002d010100050000001402d005541a050000001302d0053c1b050000001402e803341a0500
00001302e8035c1b040000002d0100000500000014020002f61705000000130200027c1b040000002d010100050000001402d005ae23050000001302d0059624050000001402e803a021050000001302e803b624040000002d0100000500000014020002621f0500000013020002d62415000000fb0280fe000000000000
9001000000000402001054696d6573204e657720526f6d616e000083040000002d01020008000000320a5a07c82301000000340008000000320a4005c22301000000310008000000320a3006902101000000310008000000320a5803cb2201000000310008000000320a4804701f01000000330008000000320a7001bc21
01000000310008000000320a6002261d01000000320008000000320a5a076e1a01000000340008000000320a40056b1a01000000350008000000320a5803681a01000000310008000000320a4804041801000000330008000000320a7001591901000000310008000000320a6002ba1501000000320008000000320a7205
1f1301000000350008000000320a5803221301000000340008000000320a4804d81001000000330008000000320a70010d1201000000310008000000320a60028e0e01000000320008000000320a7205a80b01000000350008000000320a5803360b02000000313908000000320a7001a50b01000000310008000000320a
6002ea0801000000320008000000320a8a03b20502000000313908000000320a7001240601000000350008000000320a6002860301000000320008000000320a8a033f0002000000313908000000320a70015a0002000000343310000000fb0280fe0000000000009001000000020002001053796d626f6c000204000000
2d01030004000000f001020008000000320a30067e22010000002b0008000000320a48047020010000002b0008000000320a6002321e010000002b0008000000320a6002ee1b010000003d0008000000320a48040419010000002b0008000000320a6002c616010000002b0008000000320a60028214010000003d000800
0000320a4804d811010000002b0008000000320a60029a0f010000002b0008000000320a6002560d010000003d0008000000320a6002f609010000002b0008000000320a6002b207010000003d0008000000320a60029204010000002b0008000000320a60024e02010000003d000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000ee0102022253797374656d00ee040000002d01020004000000f00103000300000000000000}{\result {\dn84 {\pict{\*\picprop\shplid1025{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}
{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pictureGray}{\sv 0}}{\sp{\sn pictureBiLevel}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn fHitTestFill}{\sv 1}}{\sp{\sn fillShape}{\sv 1}}{\sp{\sn fillUseRect}{\sv 0}}{\sp{\sn fNoFillHitTest}{\sv 0}}{\sp{\sn fLine}{\sv 0}}}
\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw10478\pich2150\picwgoal5940\pichgoal1219\wmetafile8\bliptag504379189\blipupi414{\*\blipuid 1e103735c7063503fa0b65fad26c48cd}
010009000003930200000400150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000
050000000b0200000000050000000c02a00720251200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0240000560700000b00000026060f00
0c004d617468547970650000a00209000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002dc0105000000
14020002c2050500000013020002400709000000fa02000008000000000000002200040000002d010100050000001402e803460b050000001302e803c40c0400
00002d0100000500000014020002260b0500000013020002e40c040000002d010100050000001402e8030813050000001302e803f013040000002d0100000500
000014020002ca1005000000130200021014040000002d010100050000001402d005541a050000001302d0053c1b050000001402e803341a050000001302e803
5c1b040000002d0100000500000014020002f61705000000130200027c1b040000002d010100050000001402d005ae23050000001302d0059624050000001402
e803a021050000001302e803b624040000002d0100000500000014020002621f0500000013020002d62415000000fb0280fe0000000000009001000000000402
001054696d6573204e657720526f6d616e000083040000002d01020008000000320a5a07c82301000000340008000000320a4005c22301000000310008000000
320a3006902101000000310008000000320a5803cb2201000000310008000000320a4804701f01000000330008000000320a7001bc2101000000310008000000
320a6002261d01000000320008000000320a5a076e1a01000000340008000000320a40056b1a01000000350008000000320a5803681a01000000310008000000
320a4804041801000000330008000000320a7001591901000000310008000000320a6002ba1501000000320008000000320a72051f1301000000350008000000
320a5803221301000000340008000000320a4804d81001000000330008000000320a70010d1201000000310008000000320a60028e0e01000000320008000000
320a7205a80b01000000350008000000320a5803360b02000000313908000000320a7001a50b01000000310008000000320a6002ea0801000000320008000000
320a8a03b20502000000313908000000320a7001240601000000350008000000320a6002860301000000320008000000320a8a033f0002000000313908000000
320a70015a0002000000343310000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01030004000000f001020008000000
320a30067e22010000002b0008000000320a48047020010000002b0008000000320a6002321e010000002b0008000000320a6002ee1b010000003d0008000000
320a48040419010000002b0008000000320a6002c616010000002b0008000000320a60028214010000003d0008000000320a4804d811010000002b0008000000
320a60029a0f010000002b0008000000320a6002560d010000003d0008000000320a6002f609010000002b0008000000320a6002b207010000003d0008000000
320a60029204010000002b0008000000320a60024e02010000003d000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000ee0102022253797374656d00ee040000002d01020004000000f0010300030000000000}}}}}{
\par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8192\dptxbx{\dptxbxtext\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid {\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1048\cgrid {\object\objemb\objw5941\objh1219
{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e33000000000000000000000e0000
d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffdffffff04000000feffffff05000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c0000000000000460000000000000000000000004010
31b14dc9be0103000000800200000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
0000000000000000000000001400000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002010100000003000000ffffffff0000000000000000000000000000000000000000000000000000
0000000000000000000001000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff04000000ffffffff000000000000000000000000000000000000000000000000
000000000000000000000000030000000400000000000000feffffff02000000fefffffffeffffff0500000006000000070000000800000009000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c0000000000000461700
00004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000001c000000020032d72c01000000000000d06b4900c47649000000000003010103000a01030e000001028834000288330000010288310002883900000002863d000288320002862b00030e00000102883500000102883100028839
00000002863d000288320002862b00030e000001028831000001030e00000102883100028839000001028835000000000002863d000288320002862b00030e0000010288310000010288330002862b00030e000001028834000001028835000000000002863d000288320002862b00030e00000102883100000102883300
02862b00030e000001028831000001030e0000010288350000010288340000000000000002863d004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffffffffffffffff000000000000
0000000000000000000000000000000000000000000000000000000000000400000048010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff00000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff0000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffff
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000288320002862b00030e0000010288310000010288330002862b00030e0000010288310000010288310002862b00030e000001028831000001028834000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001050000050000000d0000004d45544146494c455049435400ee2800009af7ffff300500000800ee2866080000
010009000003930200000400150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02a00720251200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0240000560700000b0000002606
0f000c004d617468547970650000a00209000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002dc010500000014020002c2050500000013020002400709000000fa02000008000000000000002200040000002d010100050000001402e803460b050000001302e803
c40c040000002d0100000500000014020002260b0500000013020002e40c040000002d010100050000001402e8030813050000001302e803f013040000002d0100000500000014020002ca1005000000130200021014040000002d010100050000001402d005541a050000001302d0053c1b050000001402e803341a0500
00001302e8035c1b040000002d0100000500000014020002f61705000000130200027c1b040000002d010100050000001402d005ae23050000001302d0059624050000001402e803a021050000001302e803b624040000002d0100000500000014020002621f0500000013020002d62415000000fb0280fe000000000000
9001000000000402001054696d6573204e657720526f6d616e000083040000002d01020008000000320a5a07c82301000000340008000000320a4005c22301000000310008000000320a3006902101000000310008000000320a5803cb2201000000310008000000320a4804701f01000000330008000000320a7001bc21
01000000310008000000320a6002261d01000000320008000000320a5a076e1a01000000340008000000320a40056b1a01000000350008000000320a5803681a01000000310008000000320a4804041801000000330008000000320a7001591901000000310008000000320a6002ba1501000000320008000000320a7205
1f1301000000350008000000320a5803221301000000340008000000320a4804d81001000000330008000000320a70010d1201000000310008000000320a60028e0e01000000320008000000320a7205a80b01000000350008000000320a5803360b02000000313908000000320a7001a50b01000000310008000000320a
6002ea0801000000320008000000320a8a03b20502000000313908000000320a7001240601000000350008000000320a6002860301000000320008000000320a8a033f0002000000313908000000320a70015a0002000000343310000000fb0280fe0000000000009001000000020002001053796d626f6c000204000000
2d01030004000000f001020008000000320a30067e22010000002b0008000000320a48047020010000002b0008000000320a6002321e010000002b0008000000320a6002ee1b010000003d0008000000320a48040419010000002b0008000000320a6002c616010000002b0008000000320a60028214010000003d000800
0000320a4804d811010000002b0008000000320a60029a0f010000002b0008000000320a6002560d010000003d0008000000320a6002f609010000002b0008000000320a6002b207010000003d0008000000320a60029204010000002b0008000000320a60024e02010000003d000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000ee0102022253797374656d00ee040000002d01020004000000f00103000300000000000000}{\result {\dn84 {\pict{\*\picprop\shplid1025{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}
{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pictureGray}{\sv 0}}{\sp{\sn pictureBiLevel}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn fHitTestFill}{\sv 1}}{\sp{\sn fillShape}{\sv 1}}{\sp{\sn fillUseRect}{\sv 0}}{\sp{\sn fNoFillHitTest}{\sv 0}}{\sp{\sn fLine}{\sv 0}}}
\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw10478\pich2150\picwgoal5940\pichgoal1219\wmetafile8\bliptag504379189\blipupi414{\*\blipuid 1e103735c7063503fa0b65fad26c48cd}
010009000003930200000400150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000
050000000b0200000000050000000c02a00720251200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0240000560700000b00000026060f00
0c004d617468547970650000a00209000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002dc0105000000
14020002c2050500000013020002400709000000fa02000008000000000000002200040000002d010100050000001402e803460b050000001302e803c40c0400
00002d0100000500000014020002260b0500000013020002e40c040000002d010100050000001402e8030813050000001302e803f013040000002d0100000500
000014020002ca1005000000130200021014040000002d010100050000001402d005541a050000001302d0053c1b050000001402e803341a050000001302e803
5c1b040000002d0100000500000014020002f61705000000130200027c1b040000002d010100050000001402d005ae23050000001302d0059624050000001402
e803a021050000001302e803b624040000002d0100000500000014020002621f0500000013020002d62415000000fb0280fe0000000000009001000000000402
001054696d6573204e657720526f6d616e000083040000002d01020008000000320a5a07c82301000000340008000000320a4005c22301000000310008000000
320a3006902101000000310008000000320a5803cb2201000000310008000000320a4804701f01000000330008000000320a7001bc2101000000310008000000
320a6002261d01000000320008000000320a5a076e1a01000000340008000000320a40056b1a01000000350008000000320a5803681a01000000310008000000
320a4804041801000000330008000000320a7001591901000000310008000000320a6002ba1501000000320008000000320a72051f1301000000350008000000
320a5803221301000000340008000000320a4804d81001000000330008000000320a70010d1201000000310008000000320a60028e0e01000000320008000000
320a7205a80b01000000350008000000320a5803360b02000000313908000000320a7001a50b01000000310008000000320a6002ea0801000000320008000000
320a8a03b20502000000313908000000320a7001240601000000350008000000320a6002860301000000320008000000320a8a033f0002000000313908000000
320a70015a0002000000343310000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01030004000000f001020008000000
320a30067e22010000002b0008000000320a48047020010000002b0008000000320a6002321e010000002b0008000000320a6002ee1b010000003d0008000000
320a48040419010000002b0008000000320a6002c616010000002b0008000000320a60028214010000003d0008000000320a4804d811010000002b0008000000
320a60029a0f010000002b0008000000320a6002560d010000003d0008000000320a6002f609010000002b0008000000320a6002b207010000003d0008000000
320a60029204010000002b0008000000320a60024e02010000003d000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000ee0102022253797374656d00ee040000002d01020004000000f0010300030000000000}}}}}{
\par }}\dpx792\dpy63\dpxsize5940\dpysize1219\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat0\dplinehollow}}}}{
\par }{\f70 Din frac\'feia 9/4 se ob\'feine }{\i x}{\i\sub 0}{\i =4*2=8, y}{\i\sub 0}{\i =-9*2=-18}{\f70 . Astfel, solu\'feia general\'e3 se poate scrie de forma: }{\i\f70 x=8+19t \'bai y=-18-43t, }{unde }{\i t}{\f70  este un num\'e3r oarecare.
\par }{
\par }{\f19\fs24\ul Implementare
\par }{
\par }{\f70 Algoritmul de mai sus este valabil, dup\'e3 cum am precizat, \'een cazul c\'e2nd cele 2 numere }{\i a}{\f70  \'bai }{\i b}{ sunt prime \'eentre ele. Dac}{\f70 \'e3 dorim rezolvarea unei ecua\'feii \'een care cele 2 numere nu sunt neap\'e3rat prime 
\'eentre ele, se poate proceda \'een felul urm\'e3tor: se calculeaz\'e3 cel mai mare divizor comun al lor (sigur este diferit de 1), iar apoi se evalueaz\'e3 dac\'e3 ecua\'feie poate sau nu avea solu\'feii, \'een func\'feie de valoarea lui }{\i c}{\f70 
. Dac\'e3 }{\i c}{\f70  este divizibil cu cmmdc-ul celor 2 numere, atunci se simplific\'e3 \'eentreaga ecua\'feie cu cmmdc \'bai problema se reduce la cea prezentat\'e3 mai sus. Dac\'e3 }{\i c}{\f70  nu se \'eemparte exact la cmmdc, atunci putem spune c
\'e3 ecua\'feia nu are solu\'feii \'eentregi.
\par }{
\par }{\f70 Pe l\'e2ng\'e3 aceasta, intervin o serie de cazuri critice \'een care algoritmul de mai sus nu poate fi aplicat, cum ar fi de exemplu cazurile \'een care nu exist\'e3 penultima frac\'feie. Dar se poate calcula solu\'feia \'bai \'een aceste cazuri:

\par }{
\par {\pntext\pard\plain\s15 \f3\fs20\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \s15\qj\fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls3\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls3\adjustright {\i 
a=0, b=0}{
\par }\pard \s15\qj\li360\nowidctlpar\widctlpar\tx392\adjustright {\'cen acest }{\f70 caz solu\'feia depinde valoarea lui c:
\par {\pntext\pard\plain\s15 \f14\fs20\lang1048\cgrid \loch\af14\dbch\af0\hich\f14 \'a7\tab}}\pard \s15\qj\fi-360\li720\nowidctlpar\widctlpar\tx392\jclisttab\tx720{\*\pn \pnlvlblt\ilvl0\ls7\pnrnot0\pnf14\pnstart1\pnindent360\pnhang{\pntxtb \'a7}}
\ls7\adjustright {c=0   =>   }{\i x }{\f70 \'bai }{\i y}{\f70  poate fi orice num\'e3r \'eentreg
\par {\pntext\pard\plain\s15 \f14\fs20\lang1048\cgrid \loch\af14\dbch\af0\hich\f14 \'a7\tab}}\pard \s15\qj\fi-360\li720\nowidctlpar\widctlpar\tx392\jclisttab\tx720{\*\pn \pnlvlblt\ilvl0\ls7\pnrnot0\pnf14\pnstart1\pnindent360\pnhang{\pntxtb \'a7}}
\ls7\adjustright {\i c }{\i {\field{\*\fldinst SYMBOL 185 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i  0 =>  }{\f70 nu exist\'e3 solu\'feii
\par {\pntext\pard\plain\s15 \f3\fs20\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \s15\qj\fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls3\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls3\adjustright {\i 
a=0, b}{\i {\field{\*\fldinst SYMBOL 185 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i  0}{
\par }\pard \s15\qj\li360\nowidctlpar\widctlpar\tx364\adjustright {\f70 Ecua\'feia devine: }{\i by=c}{. Deci }{\i y}{\f70  se poate calcula, \'fein\'e2nd \'eens\'e3 cont c\'e3 vorbim numai de numere \'eentregi.
\par {\pntext\pard\plain\s15 \f3\fs20\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \s15\qj\fi-360\li360\nowidctlpar\widctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls5\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls5\adjustright {\i a}{
\i {\field{\*\fldinst SYMBOL 185 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\i 0, b= 0}{
\par }\pard \s15\qj\li360\nowidctlpar\widctlpar\adjustright {Analog cu cazul anterior.
\par 
\par }{\f70 Dac\'e3 unul dintre numerele }{\i a}{ sau }{\i b}{\f70  are valoarea 1 nu se mai poate vorbi de penultima frac\'feie de aproximare, deci \'bai aceste cazuri trebuie tratate separat.
\par }{
\par }{\f70 O alt\'e3 observa\'feie este aceea c\'e3 frac\'feia de aproximare (}{\i m/n}{\f70 ) aproximeaz\'e3 frac\'feia (}{\i a/b}{) \'een plus sau \'een }{\f70 minus. De aceea \'een la sf\'e2r\'bait trebuie s\'e3 corectez rezultatul \'een func\'fe
ie de aceasta, \'fein\'e2nd seama \'bai de semnul frac\'feiei }{\i a/b.}{
\par }\pard \s15\qj\nowidctlpar\widctlpar\adjustright {\page }{\f19\fs24\ul Sursa programului
\par 
\par }\pard\plain \s18\nowidctlpar\widctlpar\adjustright \f2\fs20\lang1048\cgrid {\fs18 #include <stdio.h>
\par #include <conio.h>
\par #include <math.h>
\par 
\par long int v[100];
\par 
\par //obtine penultima fractie de aproximare
\par void get_mn(long int &a,long int &b,int k)
\par \{
\par  if (k>0) \{ long int aux=v[k-1]*b+a;
\par             a=b;b=aux;
\par             get_mn(a,b,k-1);
\par           \}
\par      else a=a+b-(b=a);
\par \}
\par 
\par long int _cmmdc(long int a,long int b)
\par \{
\par  while (a!=b)
\par   if (a>b) a-=b;
\par       else b-=a;
\par  return a;
\par \}
\par 
\par void stop()
\par \{
\par  printf("Nu exista solutii...\\n");
\par \}
\par 
\par int solutii(long int a,long int b,long int c,
\par             long int &x0,long int &n1,long int &y0,long int &n2)
\par \{
\par  long int m,n,cmmdc=1,_a,_b;
\par  int nr=-1;
\par 
\par  _a=labs(a);_b=labs(b);
\par  if ((_a>1)&&(_b>1)) cmmdc=_cmmdc(_a,_b);
\par  if (cmmdc!=1)\{
\par    if (c%cmmdc) \{stop();return 0;\}
\par            else a/=cmmdc,b/=cmmdc,c/=cmmdc;
\par   \}
\par  m=a;n=b;
\par  if (!(a*b))
\par   if (!a) if (!b)
\par   \{
\par    //0=c
\par     if (!c) printf("x,y}{\fs18 {\field{\*\fldinst SYMBOL 206 \\f "Symbol" \\s 9}{\fldrslt\f3\fs18}}}{\fs18 Z\\n");
\par        else stop();
\par     return 0;
\par   \}
\par    else
\par    \{ // by=c
\par      if (c%b) stop();
\par          else \{ printf("x}{\fs18 {\field{\*\fldinst SYMBOL 206 \\f "Symbol" \\s 9}{\fldrslt\f3\fs18}}}{\fs18 Z\\n");
\par                 printf("y=%ld\\n",c/b);
\par                \}
\par      return 0;
\par    \}
\par    else
\par    \{ //ax=c
\par      if (c%a) stop();
\par          else \{printf("x=%ld\\n",c/a);
\par                printf("y}{\fs18 {\field{\*\fldinst SYMBOL 206 \\f "Symbol" \\s 9}{\fldrslt\f3\fs18}}}{\fs18 Z\\n");
\par               \}
\par      return 0;
\par    \}
\par 
\par  if (labs(m)==1)
\par    \{ // inseamna ca este de forma }{\fs18 {\field{\*\fldinst SYMBOL 177 \\f "Symbol" \\s 9}{\fldrslt\f3\fs18}}}{\fs18 x+b*y=c, deci nu exista
\par      // penultima fractie de aproximare
\par      x0=c*m;n1=-b*m;
\par      y0=0;n2=1;
\par      return 1;
\par     \}
\par  else
\par  if (labs(n)==1)
\par    \{ // inseamna ca este de forma a*x}{\fs18 {\field{\*\fldinst SYMBOL 177 \\f "Symbol" \\s 9}{\fldrslt\f3\fs18}}}{\fs18 y=c, deci nu exista
\par      // penultima fractie de aproximare
\par      x0=0;n1=1;
\par      y0=c*n;n2=-a*n;
\par      return 1;
\par     \}
\par 
\par  // m si n sunt diferite de 1, si diferite intre ele
\par  m=labs(m);n=labs(n);
\par  while (m!=1)\{
\par     if (m>n)\{ v[++nr]=m/n;
\par               m-=m/n*n;
\par              \}
\par     else if (m<n) m=m+n-(n=m);
\par  \}
\par 
\par  n=v[nr];
\par  get_mn(m,n,nr);
\par  if (_a<_b) m=m+n-(n=m);
\par 
\par  //in functie de fractia de aproximare corectez rezultatul
\par  if (a*b>0) if (_a*n<_b*m) c=-c;
\par  if (a*b<0) if (_a*n>_b*m) c=-c;
\par  if (a<0) m=-m;
\par  if (b<0) n=-n;
\par 
\par  x0=n*c;n1=b;
\par  y0=-m*c;n2=-a;
\par  return 1;
\par \}
\par 
\par void main(void)
\par \{
\par 
\par  long int a,b,c,x0,y0,n1,n2;
\par 
\par  do\{
\par  clrscr();
\par  puts("Se rezolva ecuatia: ax+by=c, a,b,x,y,c\'eeZ");
\par  printf("a=");scanf("%ld",&a);fflush(stdin);
\par  printf("b=");scanf("%ld",&b);fflush(stdin);
\par  printf("c=");scanf("%ld",&c);fflush(stdin);
\par  printf("----------------------------------------\\n");
\par 
\par  if (solutii(a,b,c,x0,n1,y0,n2))\{
\par  if (!x0) if (labs(n1)!=1) printf("x=%ldt\\n",n1);
\par                      else printf("x=%st\\n",n1<0?"-":"");
\par  else if (labs(n1)!=1) printf("x=%ld%+ldt\\n",x0,n1);
\par                      else printf("x=%ld%st\\n",x0,n1<0?"-":"");
\par 
\par  if (!y0) if (labs(n2)!=1) printf("y=%ldt\\n",n2);
\par                      else printf("y=%st\\n",n2<0?"-":"");
\par  else if (labs(n2)!=1) printf("y=%ld%+ldt\\n",y0,n2);
\par                      else printf("y=%ld%st\\n",y0,n2<0?"-":"");
\par 
\par  printf("unde t este un numar intreg\\n");\}
\par  puts("Apasati o tasta (x pentru a termina)...");
\par  c=getch();
\par  \}while (c!='x');
\par }{\}
\par }}