M3P14 EXAMPLE SHEET 1 SOLUTIONS
|
|
- Paulina Watkins
- 6 years ago
- Views:
Transcription
1 M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d such tht (, b) = m + b. The d(, b) = md + db. Sice (d, db) divides both d d db, d d(, b) is lier combitio of d d db, we hve (d, db) divides d(, b). Thus (d, db) d d(, b) re equl. 1b. Let,, b be itegers d suppose tht divides, b. Show tht (,) (,) divides b. Let = (,), d =. Both re itegers. Sice divides b, we hve k = b for some iteger k, d dividig by (, ) we fid tht k = b; tht is, divides b. But (, ) = ((, ), (, ) ) = (, )(, ) by prt 1, so (, ) = 1. Thus divides b by result from lecture. 2. Express 18 s iteger lier combitio of 327 d 120. We hve: 327 = 2(120) = 1(87) = 2(33) = 1(21) = 1(12) = 1(9) + 3 Therefore: 3 = 12 9 = 12 (21 12) = 2(12) 21 = 2(33 21) 21 = 2(33) 3(21) = 2(33) 3(87 2(33)) = 8(33) 3(87) = 8(120 87) 3(87) = 8(120) 11(87) = 8(120) 11(327 2(120)) = 30(120) 11(327) 1
2 2 M3P14 EXAMPLE SHEET 1 SOLUTIONS We thus hve 18 = 180(120) 66(327). 2b. Fid, with proof, ll solutios to the lier diophtie equtio 110x + 68y = 14. We hve: 110 = 1(68) = 1(42) = 1(26) = 1(16) = 1(10) = 1(6) = 1(4) + 2 We hve (110, 68) = 2; uwidig the bove we fid 2 = 21(110) + 34(68). So 14 = 147(110) + 238(68) d oe solutio is x = 147, y = 238. The remiig solutios hve the form , for y iteger. 2c. Fid multiplictive iverse of 31 modulo 132. We must write 1 s lier combitio of 31 d 132. By Euclid s lgorithm, 132 = 4(31) = 3(8) = Thus: 1 = 8 7 = 8 (31 3(8)) = 4(8) 31 = 4(132 4(31)) 31 = 4(132) 17(31). I prticulr 17(31) is cogruet to 1 mod 132, so 17 is the desired iverse. 2d. Fid iteger cogruet to 3 mod 9 d cogruet to 1 mod 49. Note tht 1 = 11(9) 2(49). Thus 98 is cogruet to 1 mod 9 d 0 mod 49, d 99 is cogruet to 0 mod 9 d 1 mod 49. Thus 3( 98)+99 = 195 is cogruet to 3 mod 9 d 1 mod 49. 2e. Fid, with proof, the smllest oegtive iteger such tht 1 (mod 3), 4 (mod 5), d 3 (mod 7). We hve 1 = 2(3) 5, so 4(2(3)) 5 = 19 is cogruet to 1 mod 3 d 4 mod 5. So is 4. We thus kow tht 4 (mod 15) by the uiqueess i the Chiese Remider Theorem. We hve 1 = 15 2(7), so 2(4(7)) + 3(15) = 11 is cogruet to 4 (mod 15) d 3 (mod 7). Thus is cogruet to 11 (mod 105) by the Chiese remider theorem. Sice 94 is the smllest positive iteger cogruet to 11 mod 105, we must hve = Let m d be itegers. Show tht the gretest commod divisor of m d is the uique positive iteger d such tht:
3 M3P14 EXAMPLE SHEET 1 SOLUTIONS 3 d divides both m d if x divides both m d, the x divides d. (I other rigs, we will tke these properties to be the defiitio of gretest commo divisor.) It is cler tht (m, ) divides both m d. Coversely, suppose x divides both m d. Writig (m, ) = m + b for some itegers d b we see tht x divides (m, ) s well. For uiqueess, suppose there were other iteger d with these properties. The d divides m d, so it divides (m, ). Similrly, (m, ) divides m d so it divides d. Thus d = ±(m, ), d if d is positive it must equl (m, ). 4. Lest Commo Multiples 4. Let d b be ozero itegers. Show tht there is uique positive iteger m with the followig two properties: d b divide m, d If is y umber divisible by both d b, the m. The umber m is clled the lest commo multiple of d b. We first show tht m is uique. Let be other positive iteger with the sme properties. The d b divide by the first property (of ), so by the secod property (of m) we hve m. Similrly, by the first property (of m) d the secod property (of ) we hve m. So m = ±, but both re positive, so m =. Existece of m will follow from prt 2b, below: 4b. Show tht the lest commo multiple of d b is give by b. It is cler tht b b is positive. It is divisible by becuse is iteger, d similrly is divisible by b. We must thus show tht if d b divide, the so does b. If, b divide, the d b divide. But d b b re reltively prime, so their product divides 2. Thus b divides. 5. Let m d be positive itegers, d let K be the kerel of the mp: Z/mZ Z/mZ Z/Z tht tkes clss mod m to the correspodig clsses modulo m d. Show tht K hs (m, ) elemets. Wht re they? If lies i K, the 0 (mod m) d 0 (mod ). Thus both m d divide, so by questio 4, the lest commo multiple d = m (m,) of m d m divides. O the other hd it is cler tht y multiple of (m,) lies i K. Thus K cosists precisely of these multiples. Modulo m these multiples fll ito the clsses [d], [2d],... [d(m, )] = [0]. There re thus (m, ) such clsses, so K hs (m, ) elemets s climed. 6. Show tht the equtio x b (mod ) hs o solutios if b is ot divisible by (, ), d exctly (, ) solutios i Z/ otherwise.
4 4 M3P14 EXAMPLE SHEET 1 SOLUTIONS If x b (mod ), the x b = k for some iteger k. Sice both x d k re divisible by (, ), b must be s well. Suppose tht b is divisible by (, ). The solvig x b = k is the sme s solvig cx d = km, where c = (,), d = b (,), m = (,). Sice c d m re reltively prime, this equtio hs uique solutio y modulo m. Thus y x tht is cogruet to y modulo m is solutio. Sice there re (, ) cogruece clses modulo tht re cogruet to give cogruece clss mod m, we re doe. 7. For positive iteger, let σ() deote the sum d,d>0 d of the positive divisors of. Show tht the fuctio σ() is multiplictive. We must show tht σ(m) = σ()σ(m) wheever m d re reltively prime. Let d be divisor of m, d set e = (d, m), f = d e. The e d f re itegers, d e divides m. Sice (e, ) = (d, m, ) = 1, e d re reltively prime. Thus f divides by 1b bove, d f d m re reltively prime. The mp tht tkes divisor d to the correspodig e, f is bijectio betwee divisors of m d pirs cosistig of divisor of m d divisor of. We thus hve: d = ef. d m,d>0 e m,e>0 f,f>0 Fctorig the right-hd side, we fid tht it is equl to the product: e f = σ(m)σ() s climed. e m,e>0 f,f>0 8. Let p be prime, d be y iteger. Show tht p2 +p+1 is cogruet to 3 modulo p. Note tht if p does ot divide, the p 1 1 (mod p), so p (mod p). If p does divide, the d p re both 0 (mod p). Thus p (mod p) for ll. Now p2 = ( p ) p p (mod p). Thus p2 +p+1 = p2 p 3 (mod p). 9. Let be squrefree positive iteger, d suppose tht for ll primes p dividig, we hve (p 1) ( 1). Show tht for ll itegers with (, ) = 1, we hve (mod ). Sice is squrefree, it is the product of the primes tht divide it. Therefore, by the uiqueess prt of the Chiese remider theorem, it suffices to show tht (mod p) for ll p dividig. Sice (, ) = 1, p does ot divide, d so p 1 1 (mod p). Sice (p 1) ( 1), we hve = 1 + k(p 1) for some k. The = ( p 1 ) k (mod p) s climed.
5 M3P14 EXAMPLE SHEET 1 SOLUTIONS Let be positive iteger. Show tht the sum d,d>0 Φ(d) is equl to. For y positive iteger less th or equl to, (, ) is positive divisor d of. It follows (sice is the umber of positive itegers less th or equl to ) tht we hve: = #{ : 1, (, ) = d }. d,d>0 Therefore, it suffices to show tht the umber of positive itegers betwee 1 d with (, d) = d is Φ(d). But multiplictio by d gives bijectio betwee the set of positive itegers b with 1 b d d (b, d) = 1, d the set of positive itegers with 1 d (, ) = d. Thus these sets both hve Φ(d) elemets. NOTE: this c lso be doe by iductio o the umber of primes dividig.
The Structure of Z p when p is Prime
LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z
More informationNumbers (Part I) -- Solutions
Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict
More informationMATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE
MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE Prt 1. Let be odd rime d let Z such tht gcd(, 1. Show tht if is qudrtic residue mod, the is qudrtic residue mod for y ositive iteger.
More information(II.G) PRIME POWER MODULI AND POWER RESIDUES
II.G PRIME POWER MODULI AND POWER RESIDUES I II.C, we used the Chiese Remider Theorem to reduce cogrueces modulo m r i i to cogrueces modulo r i i. For exmles d roblems, we stuck with r i 1 becuse we hd
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationSimilar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication
Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationLesson 4 Linear Algebra
Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,
More informationAdvanced Algorithmic Problem Solving Le 6 Math and Search
Advced Algorithmic Prolem Solvig Le Mth d Serch Fredrik Heitz Dept of Computer d Iformtio Sciece Liköpig Uiversity Outlie Arithmetic (l. d.) Solvig lier equtio systems (l. d.) Chiese remider theorem (l.5
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationLesson-2 PROGRESSIONS AND SERIES
Lesso- PROGRESSIONS AND SERIES Arithmetic Progressio: A sequece of terms is sid to be i rithmetic progressio (A.P) whe the differece betwee y term d its preceedig term is fixed costt. This costt is clled
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationTHEORY OF EQUATIONS SYNOPSIS. Polyomil Fuctio: If,, re rel d is positive iteger, the f)x) = + x + x +.. + x is clled polyomil fuctio.. Degree of the Polyomil: The highest power of x for which the coefficiet
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationRepeated Root and Common Root
Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationStatistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006
Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationThe Elementary Arithmetic Operators of Continued Fraction
Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN 88-6785 IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr.05.0.5.697 The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationFast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation
Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777
More informationDETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1
NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More informationALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
More informationSection 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and
Sec. 7., Boyce & DiPrim, p. Sectio 7., Systems of Lier Algeric Equtios; Lier Idepedece, Eigevlues, Eigevectors I. Systems of Lier Algeric Equtios.. We c represet the system...... usig mtrices d vectors
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationInternational Baccalaureate LECTURE NOTES MATHEMATICS HL FURTHER MATHEMATICS HL Christos Nikolaidis TOPIC NUMBER THEORY
Iteratioal Baccalaureate LECTURE NOTES MATHEMATICS HL FURTHER MATHEMATICS HL TOPIC NUMBER THEORY METHODS OF PROOF. Couterexample - Cotradictio - Pigeohole Priciple Strog mathematical iductio 2 DIVISIBILITY....
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
More informationPROBLEM SET 5 SOLUTIONS. Solution. We prove that the given congruence equation has no solutions. Suppose for contradiction that. (x 2) 2 1 (mod 7).
PROBLEM SET 5 SOLUTIONS 1 Fid every iteger solutio to x 17x 5 0 mod 45 Solutio We rove that the give cogruece equatio has o solutios Suose for cotradictio that the equatio x 17x 5 0 mod 45 has a solutio
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationFermat s Little Theorem. mod 13 = 0, = }{{} mod 13 = 0. = a a a }{{} mod 13 = a 12 mod 13 = 1, mod 13 = a 13 mod 13 = a.
Departmet of Mathematical Scieces Istructor: Daiva Puciskaite Discrete Mathematics Fermat s Little Theorem 43.. For all a Z 3, calculate a 2 ad a 3. Case a = 0. 0 0 2-times Case a 0. 0 0 3-times a a 2-times
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationLecture 14. Encryption
Lecture 4. Ecryptio T. H. Corme, C. E. Leiserso d R. L. Rivest Itroductio to Algorithms, 3rd Editio, MIT Press, 2009 Sugkyukw Uiversity Hyuseug Choo choo@skku.edu Copyright 2000-207 Networkig Lbortory
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationSOLVED EXAMPLES
Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A o-zero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More information[ 47 ] then T ( m ) is true for all n a. 2. The greatest integer function : [ ] is defined by selling [ x]
[ 47 ] Number System 1. Itroductio Pricile : Let { T ( ) : N} be a set of statemets, oe for each atural umber. If (i), T ( a ) is true for some a N ad (ii) T ( k ) is true imlies T ( k 1) is true for all
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
More informationLecture 1. January 8, 2018
Lecture 1 Jauary 8, 018 1 Primes A prime umber p is a positive iteger which caot be writte as ab for some positive itegers a, b > 1. A prime p also have the property that if p ab, the p a or p b. This
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationA GENERALIZATION OF GAUSS THEOREM ON QUADRATIC FORMS
A GENERALIZATION OF GAU THEOREM ON QUADRATIC FORM Nicole I Brtu d Adi N Cret Deprtmet of Mth - Criov Uiversity, Romi ABTRACT A origil result cocerig the extesio of Guss s theorem from the theory of biry
More informationRADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals
RADICALS m 1 RADICALS Upo completio, you should be ble to defie the pricipl root of umbers simplify rdicls perform dditio, subtrctio, multiplictio, d divisio of rdicls Mthemtics Divisio, IMSP, UPLB Defiitio:
More informationTrial division, Pollard s p 1, Pollard s ρ, and Fermat s method. Christopher Koch 1. April 8, 2014
Iteger Divisio Algorithm ad Cogruece Iteger Trial divisio,,, ad with itegers mod Iverses mod Multiplicatio ad GCD Iteger Christopher Koch 1 1 Departmet of Computer Sciece ad Egieerig CSE489/589 Algorithms
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationCH 39 USING THE GCF TO REDUCE FRACTIONS
359 CH 39 USING THE GCF TO EDUCE FACTIONS educig Algeric Frctios M ost of us lered to reduce rithmetic frctio dividig the top d the ottom of the frctio the sme (o-zero) umer. For exmple, 30 30 5 75 75
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationSummary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function
Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this
More informationPrimality Test. Rong-Jaye Chen
Primality Test Rog-Jaye Che OUTLINE [1] Modular Arithmetic Algorithms [2] Quadratic Residues [3] Primality Testig p2. [1] Modular Arithmetic Algorithms 1. The itegers a divides b a b a{ 1, b} If b has
More informationPrinciples of Mathematical Analysis
Ciro Uiversity Fculty of Scieces Deprtmet of Mthemtics Priciples of Mthemticl Alysis M 232 Mostf SABRI ii Cotets Locl Study of Fuctios. Remiders......................................2 Tylor-Youg Formul..............................
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More informationICS141: Discrete Mathematics for Computer Science I
ICS4: Discrete Mthemtics for Computer Sciece I Dept. Iformtio & Computer Sci., J Stelovsky sed o slides y Dr. Bek d Dr. Still Origils y Dr. M. P. Frk d Dr. J.L. Gross Provided y McGrw-Hill 3- Quiz. gcd(84,96).
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationis an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term
Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic
More informationThe total number of permutations of S is n!. We denote the set of all permutations of S by
DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote
More informationStudent Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationNorthwest High School s Algebra 2
Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More information,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
More informationNotes 17 Sturm-Liouville Theory
ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (
More informationf(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that
Uiversity of Illiois t Ur-Chmpig Fll 6 Mth 444 Group E3 Itegrtio : correctio of the exercises.. ( Assume tht f : [, ] R is cotiuous fuctio such tht f(x for ll x (,, d f(tdt =. Show tht f(x = for ll x [,
More informationAppendix A Examples for Labs 1, 2, 3 1. FACTORING POLYNOMIALS
Appedi A Emples for Ls,,. FACTORING POLYNOMIALS Tere re m stdrd metods of fctorig tt ou ve lered i previous courses. You will uild o tese fctorig metods i our preclculus course to ele ou to fctor epressios
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationBasic Maths. Fiorella Sgallari University of Bologna, Italy Faculty of Engineering Department of Mathematics - CIRAM
Bsic Mths Fiorell Sgllri Uiversity of Bolog, Itly Fculty of Egieerig Deprtmet of Mthemtics - CIRM Mtrices Specil mtrices Lier mps Trce Determits Rk Rge Null spce Sclr products Norms Mtri orms Positive
More informationALGEBRA II CHAPTER 7 NOTES. Name
ALGEBRA II CHAPTER 7 NOTES Ne Algebr II 7. th Roots d Rtiol Expoets Tody I evlutig th roots of rel ubers usig both rdicl d rtiol expoet ottio. I successful tody whe I c evlute th roots. It is iportt for
More informationReversing the Arithmetic mean Geometric mean inequality
Reversig the Arithmetic me Geometric me iequlity Tie Lm Nguye Abstrct I this pper we discuss some iequlities which re obtied by ddig o-egtive expressio to oe of the sides of the AM-GM iequlity I this wy
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More information, we would have a series, designated as + j 1
Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite
More informationLecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,
More informationHomework 1 Solutions. The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
Homewor 1 Solutios Math 171, Sprig 2010 Hery Adams The exercises are from Foudatios of Mathematical Aalysis by Richard Johsobaugh ad W.E. Pfaffeberger. 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationEXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
More informationNotes on Dirichlet L-functions
Notes o Dirichlet L-fuctios Joth Siegel Mrch 29, 24 Cotets Beroulli Numbers d Beroulli Polyomils 2 L-fuctios 5 2. Chrcters............................... 5 2.2 Diriclet Series.............................
More information