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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## This note was uploaded on 10/21/2010 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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

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