Fall2003 Q1

Fall2003 Q1 - Page 1 of 5 Math 135 - 5.308 Aégebra Quiz 1...

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Unformatted text preview: Page 1 of 5 Math 135 - 5.308 Aégebra Quiz 1 Time: 40 minutes Sept-ember “29, 2003 NAME (undméine surname)“_mm_m__m_ - ¢._.____ N0 uaicujlmx' allowed! We at Page 2 of 5 Question 1 Let a, b E Z. [3] Define what is means for d to be the greatest common divisor of a and b. D» are. in W‘» J T951113"? \J 1 {IL \wJe .3 a » ah d de; [2] (ii) Suppose that a, b are not zeros and dla and dlb. Prove that if there exist $0, yo 6 Z Alsuch that d = a - x0 + b - yo, then (1 = GCD(a, b). " 3 v 7 El . a, _ w W ,, \ r _. New» *9 (NOV? {Wei dem 0% a QAQL k3 (Wm d \t «L i" -- t V t. t 1 . Le Cr 0Q 00:» €th Qflamozramgx g; Law” (3;; I"; ‘ ‘v-“T: Am J '7 ,4 Va “£6 “3 {J Z Q‘Ox ooh CUE “3‘3 W?- V‘HI’E‘B dcwm‘wo 1» o\ d I ‘ \K jog-9 dka and; d“) dd ,l’erq @=.fi¢¢}<q 2b) Z/ 7* U . 7/ \ ’ Page 3 of 5 \ , . M . >86. 962%“ 4ier ‘4' [5] (iii) Suppose that a, b are not zeros. We have seen in class that the converse state— 5 ment of (ii) is also true. Use these results to prove that for c E Z, if GCD(a, c) = 1, then GCD(ab, c) = GCD(b, c). I Let: Gmoiea>sa Prou? eyes. 3%» 5 ix «rs # g\~4nr r a ‘ " J‘ ‘ l {:3 E Q Q} ‘ ‘3 C; N\, (if j “A :,{ Q; ; J QED that» "o w n ESE. '4 _ h v I .\ \JL T) \_ okx’xo $30 JM’SOE’ ZZ >‘hCP E:C0(b,a: " d 5 5H”) 2") WA M \‘Q A j .;u\ __ 1 r ‘ \ “x W -‘—/ \“x_- a A 3 s4 1.“; Ewan? i'Cikxb S h I ‘j‘ \ L6 émibfi "LED-rd Us” by») A; V e a _ m i‘ " A: if. ". e\o\h arid xix/fl fink ,Ox u5\ CA“) QWQ‘: C 5 Rex 3km»: d, e 16. e: f d re 22 e {in / e : erx : I 9.0/15; +_€C%a v ” 1’ \1 (\flrw > J 5mg? 6c?) 2 r.\ = g a w) ‘N J ‘ Q ’2 2 I OM I ' b at )—E&? ’0 / E '5’? .. 4x g; '2 / I v ' .1 LC." {(7 F,..‘:. A" 5:“ ‘fémbflhco’é t\ - . a, QC _~ Czdfl 2Q??? ) eaSCdCbpb 5hov~ cue aid and (1,920 «.7 dze @4461 show die ficdmgt, 3 .403 ab/x {3:3 H.209 Slab/31' +Cb5 3b Page 4 of 5 Question 2 [6] Find the complete solution to the Diophantine equation 143cc + 1873; = 33. Q K MA) Question 3 [2] (i) Find GCD(24 - 3 - 7, 22 - 72 - 11). B5 ’(‘wm- ‘\ C/ICDC’LV‘bfl/ Q37? x,\\ 2. Q1 .7 [2] (ii) Convert (487)10 to base 7. Page 5 of 5 ...
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Fall2003 Q1 - Page 1 of 5 Math 135 - 5.308 Aégebra Quiz 1...

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