MATH135 Lecture 6: 6

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Alan Thompson

Document Summary

2 ) ( 2 , 3 ) # 1 7 , 6 = 3 ( 2 7 -1 0 a = 6 , 4. , 0 ) gcd ( 0 , x ) Proofletdigcdl3atb. at andfgcdla. by/tsd=gcdl3atb,a),weknowdl3atbanddla bydefinitonofgcd. thenbydic. weknowdlbatbmtlall-31. si nce3atb-3a-biweknowdlbthendlaanddlbmeansd. is acommondnisorofaandbasfgcdla. to wehauedef. now. asfgcdla. to flaandflblhusbydis. fi andsofl3atb. asflaandfl3atb. weknowfisa. com mondiusorofaand3atb. thusfdsincedfandfd. de. 1 3) ( a ) + 1 1 ) ( b) fandsotheshtementholdihreatestlom. mn/7iusorsrecah:d= gcdca. br ) > Idaid-gcdlx. gs gcdlaytryjfgcdlg. rs usmgdefgcdanddictogetdlrandsodtfihenusingdef. no fgcdanddicgougetflxandfly. sofd. us fdex. x-3atbiyia. r-b. q-3x-note. youhave. to dedwiththcasex-g-oseparatelgexiletabc-z. bouegcd. ba tb , a ) Bgthegcduithkemaindersheoremgcdl3atb. aj-gcdla. rs sina. b-qatr. gcdlabkgcdla. rs bygcduithkmmainderssogcdl3atb. az gcdlabltenathesestatmentholds. lt gcdla , exigcdl56. fi rgcd. 2 1 ) 3 5 = b ( 1 ) 3 5 + 2 1 ( 1) 2 1 + 1 4 ( 1 ) 1 4 t 7 ( 2 ) 7 -1. 1 7 1 = 7 gcd 156 , 3 5) = k - 1 sina. oericri. ie uclideanalgorithmhivenaandbuithbtodivideabybtog. ee tnemainderr. dividebbgr. to getremandern. 2 3 1 gcd ( 1 2 3 9 7 3 5 ) 1 2 3 9 5 + 7 3 5 y.