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UNIVERSITY WATERLOO, ONTARIO
MATHEMATICS 135 ALGEBRA FINAL EXAMINATION 12 December 2001, 9:00 a.m. - Noon Exam Type: Special Materials Special Materials Allowed: Non-programmable calculator SURNAME: (Print) Signature: INITIALS: ID: LECTURE SECTION: Instructions to Candidates TUTORIAL SECTION (if known): \/ Lec sect Instructor Lecture times 85 room Tutorial sections
I: LEC 001 S. Furino 10:30 a.m. MWF STJ 315 TLT 101 [:1 LEC 002 L. Cummings 8:30 a.m. MWF MC 2017 TLT 102, TUT 103
D LEC 003 F. Zorzitto 9:30 a.m. MWF MC 2017 TUT 104, TUT 105
El LEC 004 L. Cummings 10:30 a.m. VIWF MC 2017 TUT 106, TUT 107
D LEC 005 D. Younger 11:30 a.m. VIWF MC 4059 TLT 108, TUT 109
E] LEC 006 M. Bauer 12:30 p.m. VIWF MC 4059 TLT 110, TUT 111
I: LEC 007 D. Younger 1:30 p.m. 'VIWF MC 2017 TLT 112, TUT 113
E! LEC 008 I. VanderBurgh 2:30 p.m. MWF MC 2017 TUT 114, TUT 115
D LEC 009 E. Teske 8:30 a.m. MWF MC 2035 TLT 116, TUT 117
[:1 LEC 010 I. VanderBurgh 9:30 a.m. MWF MC 2035 TUT 118, TUT 119
D LEC 011 E. Teske 10:30 a.m. VIWF MC 2035 TUT 120, TUT 121
El LEC 012 W. Gilbert 11:30 a.m. MWF MC 4045 TUT 122, TUT 123 1. Write your name, signature, ID number, lecture and tutorial section at the top of the page, and tick your
lecture section. 2. Questions are to be answered in the space provided. Give REASONS for your answers (except in Question
10) and show all your work required to obtain your answers. If the space is insufﬁcient, use the back or the
blank page at the end, and indicate where your work continues. 3. There are 13 pages in this examination, including a blank page at the end. Check that you have all the pages.
Write your name at the top of each page. 4. Non—programmable calculators are permitted for calculation only. You should show your intermedicate steps.
Use of pre-stored information is cheating. For Marker Only 1. (a) Find the complete integer solution to the Diophantine equation 123:1: — 216g 2 39. (b) List those solutions to the Diophantine equation in part (a) for which 1* and y are both positive integers
satisfying :8 + y g 300. 2. (a) State Fermat’s Little Theorem. (b) Find all the integer solutions of the congruence x11 + 71' E 18(m0d77). 3. Solve the following system of linear congruences.
:c E 12(mod 20)
a: E 11(mod 39) 4. Let a sequence be deﬁned by $1 = 10, x2 = 16, and 1B,, : 313n_1 - 213,14 for n 2 3.
Prove that In : 4 + 3(2")
for all positive integers n. 5. (a) In an RSA cryptosystem suppose that the public key is (e, n) : (7, 143). Find the private key.
(b) Find the ciphertext when the message M = 103 is encrypted using the public (7, 143). 6. (a) Suppose a, b, c are positive integers and GC'D(b, c) = 1.
Prove: If a|5b+ c and a|6b + c then a = 1. (b) Let a and b be nonzero integers, and let (1 = GCD(a, b). Prove that GCD(%, RIO‘ )=1. 7. (a) Write the complex number — @040 in the standard form x + yi. (b) Solve the equation 2 = 22 for z E (C, and plot your solutions in the Argand diagram. (You can give your
answers in standard form or polar form). 8. (a) Give a careful statement (Without proof) of the Rational Roots Theorem.
(b) Find all the rational roots of = 4x4 — 2233 —— CL‘\— 1. 9. (a) Prove: Theorem: If c is a complex number that is a root of a polynomial f whose coefﬁcients are
real, then the complex conjugate 6 is also a root. (b) Find all the roots of = x6 + 16183 + 64 in (C.
(c) Factor f : £86 + 162:3 + 64 into irreducible polynomials in (d) Factor f = 9:6 + 167:3 + 64 into irreducible polynomials in Z7[:1:], where the coefﬁcients in f are considered as elements of Z7. 10. For each of the following pair of statements, tick in the boxes whether they are equivalent or not. In (d) and
(e) the universe of discourse is the set of real numbers. (No reasons are required.) (a) P AND Q,Q AND P (b) NOT (P AND Q), ( NOT P) AND ( NOT Q)
(c) NOT (P => Q), ( NOT P) AND Q (d) Vz3y(:c + y > 0), ElyVac(x + y > 0) (e) NOT (VrEly(at + y > 0)), Elz‘v’y(a: + y g 0); ...
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