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### math135-a5

Course: MATH 135, Winter 2011
School: Waterloo
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Word Count: 138

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5 Assignment Questions 1. Prove that any two consecutive integers are coprime. 2. Prove that the Diophantine equation ax2 +by 2 = c does not have any integer solutions unless gcd(a, b)|c. If gcd(a, b)|c, does the equation always have an integer solution? 3. (a) Prove that if n 3, then n! + 3 is not prime. (b) Prove that for every k P, k consecutive positive integers that are not prime can found. be (A good way...

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5 Assignment Questions 1. Prove that any two consecutive integers are coprime. 2. Prove that the Diophantine equation ax2 +by 2 = c does not have any integer solutions unless gcd(a, b)|c. If gcd(a, b)|c, does the equation always have an integer solution? 3. (a) Prove that if n 3, then n! + 3 is not prime. (b) Prove that for every k P, k consecutive positive integers that are not prime can found. be (A good way to prove this is to explicitly show what these k integers could be.) 4. Either prove or give a counterexample to, a3 |b3 if and only if a|b. 5. Find the prime factorization of gcd(a, b) and of lcm(a, b), where a = 25! and b = (5500)3 (1001)2 . 6. Find the number of positive integers whose prime factors are 2, 3 and 5 and which have exactly 20 positive divisors. 1
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Waterloo - MATH - 135
Assignment 6 Questions1. Suppose that a, b Z and m, n P. Prove that an bn (mod mn) if and only ifa b (mod m).2. Prove that if n 3 (mod 4) then n is not the sum of two squares.3. Find all possible pairs of digits (a, b) such that 99 38a91b.4. Let n =
Waterloo - MATH - 135
Assignment 7 Questions1. Prove, without using a calculator, that 641 divides the Fermat number5F (5) = 22 + 1.92. Find 277 (mod 11).3. Let p be a prime number and a a positive integer not divisible by p. Fermats LittleTheorem tells us ap1 1 (mod p)
Waterloo - MATH - 135
Assignment 8 Questions1. Solve the following pairs of congruences.15x 4(mod 26)24x 6(mod 63)2. Chinese generals used to count their troops by telling them to form groups of somesize n, and then counting the number of troops left over. Suppose there
Waterloo - MATH - 135
Assignment 9 Questions1. (a) Given n = pq , p &gt; q , and (n) = (p 1)(q 1) prove thatp + q = n (n) + 1andpq =(p + q )2 4n.(b) The integer n = 71531 is the product of two primes p and q , and the Euler phifunction (n) = (p 1)(q 1) = 70992. Determine t
Waterloo - MATH - 135
Assignment 11 Questions1. Express the complex number (1 83i) in standard form.2. (a) Use De Moivres Theorem to prove thatcos 4 = 8 cos4 8 cos2 + 1sin 4 = 4 cos (sin 2 sin3 ).(b) Calculate cos 4 if cos = 1/ 3.3. Find all the complex solutions to z
Waterloo - MATH - 135
Assignment 11 Solutions1. Express the complex number (1 83i) in standard form.Solution:|1 3i| = 2 and the argument of (1 3i) is = 5/3. So (1 3i) = 2(cos(5/3)+i sin(5/3). By De Moivres Theorem, we have(1 3i)8 = 28 (cos(40/3) + i sin(40/3)= 28 (cos(
Waterloo - MATH - 138
Waterloo - MATH - 138
Proposed schedule and list of recommended exercisesMath 138, Winter 2011WeekDatesSections and topicsAsstReview,pp. 410-411 # 13,17, 25, 33,Section 5.3 : Quick review of The Fundamental Thmof Calculus. (Covered in Math 137)1Jan. 5-7(Lectures 1
Waterloo - MATH - 138
Math 138, Winter 2011, Assignment 1 solutions.
Waterloo - MATH - 138
Math 138, Winter 2011, Assignment 2 solutions.
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 3Topics: Volumes by disks, volumes by cylindrical shells.Due: 11 a.m. Friday, January 28.Hand in your solutions to the following problems.1. Evaluate the following integrals:(a)1 x3 4x10dx0 x2 x63 4x10x=xx2 x63
Waterloo - MATH - 138
4.(a)2y ' = 1 xyy1The gray line is the solution curvepassing through the point (0,0)0-1-2-2(b)-1012xy ' = x xy2y1The gray line is the solution curvepassing through the point (1,0)0-1-2-2(c)-1012x2y ' = x (2 y)y1The gr
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 5 Solutions1. Find the general solutions of the following linear DEs:(a)dy 2xy = 2xdxR2The integrating factor is I (x) = e 2x dx = ex . Then multiplyingthrough by I (x) gives:dy x222e 2xyex = 2xexdxd22ex
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 6Topics: Sequences, series.Due: 11 a.m. Friday, February 18.1. Prove the following theorem in two ways: If lim |an | = 0, then lim an =nn0.(a) Using the squeeze theorem for sequences.Proof: We have |an | an |an |. W
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 7 Solutions1. Suppose cfw_an is the sequence dened recursively bya1 = 0an+1 =5 + 2anfor n = 1, 2, 3.Prove that cfw_an converges and nd its limit.The rst few terms of the sequence are: a1 = 0, a2 = 5 2.23, a3 =5 +
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 8 Solutions1. (a) We note that 0 &lt;2n3n +1&lt;2n3nn=2n3for every positive integer n.2converges sinceBy the Comparison Test,n+13n=12geometric series (with a = r = 3 ).(b) We apply the Limit Comparison Test to
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 9 Solutions1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.(1)n1(a)n=12n4We consider the series of absolute values,n=121=2, which4nn4n=1(1)n1clearly converg
Waterloo - MATH - 138
MATH 138 Winter 2011Solution: Note thatAssignment 10Topics: Representation of a function as a powerseries, Taylor and Maclaurin series.Due: 11 a.m. Friday, March 25.1x21d=3 x3 )23 x3(a3 dx a1d1=333a dx 1 x3a1. Find a power series repre
Waterloo - MATH - 138
MATH138: Exam-AIDSOSPrepared by Vincent Chanv2chan@math.uwaterloo.caApril 8, 2010I include numbers after each section heading that correspond to the textbook section headings for yourconvenience. Proofs of theorems are included in your textbook, and
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 1Topics: Integration review, integration by substitution and by parts.Due: 11 a.m. Friday, January 14.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last n
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 2Topics: Trigonometric integrals, trigonometric substitution, completing the square andtrig substitution.Due: 11 a.m. Friday, January 21.Instructions: Print your name and I.D. number at the top of the rst page of your
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 3Topics: Volumes by disks, volumes by cylindrical shells.Due: 11 a.m. Friday, January 28.Hand in your solutions to the following problems.1. Evaluate the following integrals:(a)(b)1 x3 4x10dx0 x2 x6x2 2x1dx(x1)2
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 5Topics: Linear differential equations, applications.Due: 11 a.m. Friday, February 11.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last name. Submit you
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 6Topics: Sequences, series.Due: 11 a.m. Friday, February 18.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last name. Submit your solutions in the same or
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 7Topics: Sequences, Series, the Integral Test, Estimates of sumsDue: 11 a.m. Friday, March 4.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last name. Sub
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 8Topics: Comparison test, alternating series test.Due: 11 a.m. Friday, March 11.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last name. Submit your solu
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 9Topics: Absolute Convergence, Ratio Test, Root Test, Power Series.Due: 11 a.m. Friday, March 18.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderline your last name.
Waterloo - MATH - 138
MATH 138 Winter 2011Assignment 10Topics: Representation of a function as a power series, Taylor and Maclaurinseries.Due: 11 a.m. Friday, March 25.Instructions: Print your name and I.D. number at the top of the rst page of your solutions, andunderli
Waterloo - MATH - 138
MATH 138Assignment 11NOT TO BE HANDED INOur course ends with an introduction to parametrized curves. This material iscovered in Stewart, Sections 10.1 and 10.2. Section 10.1 is intended to get youfamiliar with the notion of a parametrized curve. The
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Ancient Greek Architecture vs. Roman ArchitectureThe Ancient Greek and the Romans had several differences and similarities when itcame to architecture. Ancient Greeks typically built with mud, wood, plaster and stone.(How to compare and contrast Greek
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Film Art vs. Popular CinemaPopular cinema is a style of entertainment, usually in the form of a movie that is enjoyed by a largeportion of the population rather than just a select group. Also in the book, The Art Cinema as a Modeof Film Practice, the a
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IconographyAna GonzalezAxia CollegeArtists have four important roles.(Sanjay,2009) First, an artist can help us see thingsdifferently. They can take everyday things and make their audience see them in a differentway. Take for example Andy Warhol and
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Neoclassicism 1Neoclassicism, Impressionism, and Abstract ArtAna GonzalezAxia CollegeNeoclassicism 2Neoclassicism, Impressionism and Abstract art are very different schools of art. They portrayimages in different ways. Neoclassicism and Abstract art
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1LodgeMedicineCreekLodgeIntheartlabweareabletoobservetwodrawingsofthesameevent.Eventhoughtheyarethesamesituation,therearesomebigdifferencesthathavetodowiththeartistownbeliefsandviewofthesituationandthepeoplepresentatthetime.HowlingWolfsTreatySigninga
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DavidThere are some differences between both the three David sculptures, but the one thatstands out the most is the era it was created. Michelangelos was during the Renaissance,Berninis is Baroque, and Donatello carved his David during the Gothic perio
University of Phoenix - AED - 204
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Axia College MaterialAppendix FImpact Matrix AssignmentDirections: Assume that you are on the school board of your school district and you are tasked with the job of strategic planning forissues facing your district. Complete the matrix below to deter
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I was watching Two and a Half Men and there was a very funny scene. Alan one of the maincharacters was about to enter his son's bedroom. He is knocking on the door and there is a mediumshot. He opens the bedroom and the shot becomes a full shot of the e
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1/ Gina needs to interview many people. As a new store manager, she needsto get acquainted with her regional boss Marty, hire an assistant manager,hire two new clercks, schedule review performances, talks to formeremployees and customers as well.Gina
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Name:MaryHannahACCT4230Chapter1Date:6/6/2011ProblemI:EdwardandEmilyJonesaremarriedandfileajointreturn.Theyanticipatehavingtaxableincomeof\$240,000inthenextyearandareconsideringwhetherornottopurchaseapersonalresidence.TheresidencewouldprovideEdandE
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Running head: COMPARISON OF THREE SCULPTURESComparison of Three SculpturesSherry Donath HenningerART 101April 29, 2011Jeff Lyttle1COMPARISON OF THREESCULPTURES2Comparison of Three SculpturesDonatellos David is of a young boy, not very muscular
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1Analytical and Expressive LinesUNIVERSITY OF PHOENIX- AXIA COLLEGEAnalytical and ExpressiveLinesART101Sherry Donath Henninger3/25/20113Analytical and Expressive LinesAnalytical and Expressive LinesThe first of the two paintings that I chose be
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Unit 7- websites.http:/chemed.chem.purdue.edu/genchem/topicreview/bp/ch9/redox.phphttp:/www.wisegeek.com/what-is-carbonation.htmhttp:/www.byo.com/resources/carbonationhttp:/www.floodsmart.gov/floodsmart/
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In reading this paper, I would have most defiantly made the conclusion paragraph longer than onesentence, I mean although the word count was met it is still not done right on that note alone. Within thefirst two paragraphs there was a lot of information
University of Phoenix - COM - 156
I Renting an apartment is a better option than buying a home. Many people would think theopposite of this when it comes to our economy today.II Lets look at the statistics of renting, versus buying, shall we? The break down betweenrenting and buying is
University of Phoenix - COM - 156
Renting an apartment is a better option than buying a home. Manypeople would think the opposite of this when it comes to our economy today.Let us look at a home that is for sale at \$395,000 that is a 4 bedroom with 2.5 baths, and sitting on2 acres of l
University of Phoenix - COM - 156
Appendix BCOM/156 Version 41Axia College MaterialAppendix BSelecting a Topic for Your Final PaperThis week you choose a topic for a persuasive essay, which is sometimes called an argument.You research and develop this topic throughout this class, a
University of Phoenix - COM - 156
10. PERSUASIVE ESSAYUniversal Health Care Coverage for the U.S.The United States is the only modernized Western nation that does not offer publicallyfunded health care toall its citizens; the costs of health care for the uninsured in the United States
University of Phoenix - COM - 156
Axia College MaterialAppendix GFinal AssignmentNote that there are multiple parts for this final assignment.Read two of the five paragraphs on the second page of this appendix. Write a brief paragraphof about 150 words in which you do the following:
University of Phoenix - COM - 156
US/101 Midterm ExamUserFirst NameLast NameConfirmation #ScoreTotal Questions:Total CorrectStart:End:9019173533WHITNEYHOUSTON165523152229222/27/2011 12:22:28 PM2/27/2011 12:32:50 PMHereissomeadditionalinformationonitemsyoumissed:Topic:D
University of Phoenix - COM - 156
US/101 Midterm ExamUserFirst NameLast NameConfirmation #ScoreTotal Questions:Total CorrectStart:End:9019173533WHITNEYHOUSTON165523152229222/27/2011 12:22:28 PM2/27/2011 12:32:50 PMHereissomeadditionalinformationonitemsyoumissed:Topic:D
University of Phoenix - COM - 156
Summarize what you learned in this Orientation Workshop.o How will you apply what you learned in this Orientation Workshop to youreducational, personal, or professional pursuits?o Provide relevant personal notes and observations, such as anything discu
University of Phoenix - COM - 156
ReferencesBook Authors: Dowd K.O., Janasz S.C., and Schneider B.Z.Publisher: McGraw-HillPublication Date: 2002Book Title: Interpersonal Skills in OrganizationsCity of Publication: New York, NYPage Number: 372
University of Phoenix - COM - 156
Time LogTime6:00 a.m.6:30 a.m.7:00 a.m.7:30 a.m.8:00 a.m.8:30 a.m.9:00 a.m.9:30 a.m.10:00 a.m.10:30 a.m.11:00 a.m.11:30 a.m.12:00 p.m.12:30 p.m.1:00 p.m.ActivitiesComments or Observations1:30 p.m.2:00 p.m.2:30 p.m.3:00 p.m.3:30 p.m.
University of Phoenix - COM - 156
Axia College MaterialAppendix ACompare Online and Traditional College EducationsWhat is Online Learning?Online learning includes courses taken at a distance via the Internet with no face-to-face contactwith instructors (Doyle, 2009). There are millio
University of Phoenix - COM - 156
Assignment 1: University Library SearchT he directors at MUST University go on about how the amount of students who graduatecollege end up with a high debt, and spend most of their paychecks on paying back financial AidLoans. A point made is that the r
University of Phoenix - COM - 156
Welcome to WritePoint, the automated review system that recognizes errors mostcommonly made by university students in academic essays. The system embedscomments into your paper and suggests possible changes in grammar and style. Pleaseevaluate each com
University of Phoenix - COM - 156
I feel that I would have become persuaded if there had been more information and facts about theUHC through the world, and just how well it would affect the USA. I did like reading what informationwas there as well as the students thoughts and concerns
University of Phoenix - COM - 156
Personally I am a fan of music, not just one kind either, but all. This is something that there is muchcontroversy over, mind you there has been for many years, and I do not think it will change through timeeither. Music is a part of almost every person
University of Phoenix - COM - 156
The film Harry Potter and the Deathly Hallows Part 2 which was releases on July 15th, 2011 atMidnight is guaranteed to be yet another box office hit for many theaters through the country. Many fansof the books, as well as the movies waited long and grue