Spring2006 M

Spring2006 M - Given Name: Id. No.: , Math 135 Algebra for...

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Unformatted text preview: Given Name: Id. No.: , Math 135 Algebra for Honours Mathematics Mid-Term Examination 2006-06-05 7:00-9:00 Instructor: B. Tasic Instructions: 1. Show your work in the space provided. Be precise, concise and complete. If you need more space, use the back of the page or the blank page at the end, but state (in the space provided) where your work is continued.’ 2. This test has 6 questions. Math 135 Midterm Exam, Spring 2006 Page 2 of 8 Name: . 1. (a) Define GCD(a, b) for integers a, b. 5 if) ' I 5 wwwva M, 0m W'éw?’ 52% M g 4%; {is m ‘ A? .W mm] a W'Méfl (13) Find GCD(20!, 20062006) 1” a :2 I 2 3 2 g 2 ‘ J} 7 2 3 , g , 3'5 z :7 322 Mi: {wg Math 135 Midterm Exam, Spring 2006 Page 3 of 8 Name: 2. Let a and b be integers. Show that if GCD(a, b) = 1 then GCD(a — b,a + b) = 1 or 2. 343 64%.. [512} gr i M43 MW! ' // // S [it-1’} f Wé/mj 8»: vai a?” :c-f CW”§,Q/{é? #423“ ._ (“'922c4’(a+£75 _‘ €95 taffééwfé'm $5512? ach )wLHV'Xkali Jug-é fléQa/éwv (not! 5) M: CocJ777fHé/n’ lag ‘ m—mbfl fig“: é‘fi :3 J‘ at??? 7; «kr’ i» {E7 $7 Math 135 Midterm Exam, Spring 2006 Page 4 of 8 Name: 3. (a) Define What it means for integers a and b to be coprime. 5" 3%” 57*” ’ V ‘5 ‘ J, .r’ r (‘5 WA? flar’lf 534“ I [:7 (94,179,, “In/IprTw’ fflaflw-Ml’ fi?/m»¢ (b) Prove that if integers a and b are coprime then ab and a + b yopfime. ‘ 9/00/50“ {777/ 7”” ~ mg, ME): if ; * * flflflflflflfl “W3 1’ $3 I 7> a (w w cm + ’7 x «« Apr!" s: /‘ 3;}{zfif‘5f5fi 46921» ; a (XIWCvfléj yé/fiflga 47> 7f :— / I , ' i Math 135 Midterm Exam, Spring 2006 Page 5 of 8 Name: 4. A sporting-goods store placed a total order of $2490 for a number of bicycles at $29 each and a number at $33 each. How many bicycles of each kind were ordered? {WW—f 2,4,3”; u/‘tfl / a ’5'} 0’ : 2,01 {/y 4 ‘,%j§:§*3‘3+4 /7 g / M 19 :szl 27 /370 £959 4:441”? flewmwefi I, 7 }(’/7470)*Z"1("7920) r2 24490 7<°’*!7¢?6?+2W 2’0 art?) 447/ 45:22 V: “Way’fim $0 :7 M 5 60,; Kt“? MEWMQW? 2g: ‘/:1 9%:{2 ijfévy f: f;"[/ w 79,, {7476’ 2*" 2019(6’935’ 3'7 V; 4’9on "igféfi; 5" Z TZEfiQéZf é’l/ "ML-$4 fig 5127 Z if hiya/{25 q a?” z/ élaym/IS 207 5'7 era/v'cflfwf {7; Math 135 Midterm Exam, Spring 2006 Page 6 of 8 Name: 5. (a) Define prime number. 77 {9 I M/ #71 \ (b) Prove that the number of primes is infinite. Math 135 Midterm Exam, Spring 2006 Page 7 of 8 Name: 6. (a) Let a and b be integers. Prove that a3Ib3 if and only if alb (b) Prove or give a counter example For a, b, E Z G'CD(a2, b2) = (G'C'D(a, b))2 X? ig/Ca‘zr'é’z ‘ 360401382)“: [flcflflflflfr M 2. .15 Wiflfvé’ff , 6wan I /7 2 I Math 135 Midterm Exam, Spring 2006 //<‘ Page 8 of 8 Name: Blank page / V / x; ,_ ; .r I m/ I or m f ' ( ,\ fr .4; 0110! . 3 3' \ , a! 5, ...
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Spring2006 M - Given Name: Id. No.: , Math 135 Algebra for...

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