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Unformatted text preview: Math 135 ——— Fall 2001 —— Final Page I of I REPRINTED BY MATHSOC {>4 Instructors: M. Bauer, L. Cummings, S. Furino, W. Gilbert, E. Teske, I. VanderBurgh, D. Younger, F. Zorzitto
[10] 1. (a) Find the complete integer solution to the Diophantine equation 123x — 2163; = 39. (b) List those solutions to the Diophantine equation in part (a) for which :3 and y are both positive integers
satisfying .1: + y f 300. [10] ‘2. (a) State Fermat’s Little Theorem. (b) Find all the integer solutions of the congruence 1:11 + 7.3 E 18 (mod 77). [10] 3. Solve the following system of linear congruences. :L' 12 (mod 20)
2: : 11 (mod 39) [10] 4. Let a sequence be deﬁned by $1 = 10, $2 : 16, and 22,. = 3xn_1 ~ 211,4 for n 2 3. Prove that
xn z 4 + 3(2") for all positive integers n. U! [10] (a) In an RSA cryptosystem suppose that the public key 1s (6, n): (7 143). Find the private key. (b) Find the ciphertext when the message M = 103 is encrypted using the public key (7, 143).
[10] 6. (a) Suppose a, b, c are positive integers and GCD(I), c) = 1. Prove: If a5b + c and a6b+ c then a : 1. (b) Let a and b be nonzero integers and let (1 : GCD(a, b). Prove that GCD (g, %> = 1.
f —10
. 1 3 . .
[10] 7. (a) erte the complex number (3 — 41) in the standard form a: + yi.
(b) Solve the equation 3 : 22 for z E C. and plot your solutions in the Argand diagram. (You can give your
answers in standard form or polar form.) [10] 8 (a) Give a careful statement (without proof) of the Rational Roots Theorem.
(b) Find all the rational roots f(.1:) : 4.2:“ ~ 2.123 — :1: — 1.
[15] 9 (a) Prove: Theorem: If c is a complex number that is a root of a polynomial f(x) whose coefﬁcents are
real. then the complex conjugate E is also a root.
(b) Find all the roots of f(12) : :126 —l— 16.133 —l— 64 in C
(c) Factor f(.r) : 1’6 + 1.6.r3 + 64 into irreducible polynomials in lPZ[.r]. 1
(d) Factor f(:c) = x6 + 161:3 + 64 into irreducible polynomials in Z7[x], where the coefﬁcients in f(;1:) are
considered as elements of 27M].
[5] 10. For each of the following pairs of statements. determine whether they are equivalent or not. In ((1) and (e) the universe of discourse is the set of real numbers. (N0 reasons are required.) (a) P ANDQ QANDP (b) NOT( P AND Q) (NOT P) AND (NOT Q)
(c)1 NOT( P :> Q) (NOT P) AND Q d) V.L3y(l'+y>0) Elsz'(.L*+y>0) (e) NOT(VLII 3y (Ily >0)) 3:1: Vy (ar+y_<_ 0) Mpg—H—
DOAsub
Ham 110] [8] Math 135 — Fall 2000 —— Final Page 1 of 1 REPRINTED BY MATHSOC D<l Instructors: R. Andre, S. D’Alessio, J.Geelen, J.Hooper, L. Liptak, C.T. Ng, I. VanderBurgh, S. Wolf, D. Younger 1. Solve the following system of linear congruences: 2 4. 5 6 9 11:1: E 12 (mod 24)
a: E 4 (mod 25). . Let P and Q be statements. Deﬁne the statement P 0 Q by the following truth table: (a) Show that P 0 Q is equivalent to the statement NOT(P AND Q).
(b) Is P AND Q equivalent to (P 0 Q) 0 (Q o P)? Justify. . Let a1 : 2,, a2 = 3, and an 2 3an_1 — 2an2 for n 2 3. Show that an : 2"—1 + 1 for n 2 1. (a) Prove the following theorem: If a, b,c E Z with club and GCD(a, c) = 1, then clb.
(b) Prove: For integers a, b, c such that GCD((1, b) : 1, if ale and blc, then able.
. Consider the RSA scheme with public key (e, n) : (23,407). (a) Using the above RSA scheme. encrypt the message JVI : 321.
(b) Determine the private key corresponding to the public key (23, 407). . Determine all complex numbers (in polar or standard form) of the equation 26+2iz3—420. (a) State the Rational Roots Theorem.
(b) Let f(.l}) : 4.1:4 + 8x3 + 9:r2 + 5x + 1. Determine the rational roots of f($).
(c) Let f(.r) = 4.7:4 + 8.133 + 99:2 + 51: + 1 as in part (b). Factor f(:L‘) into irreducible factors in Qh’), Rh]. Ch]. and 35ft]. . Let j’(.r) : r3 — 3f + an + I) be a polynomial in which the coefﬁcients a. and I) are both real. If —1 + x/31 is
a root of f(;1:), determine the values of a and b and ﬁnd all the roots of f(9:). . Show that for every integer a,
(1561 E a (mod 561) even though 561 is not prime. (Hint: Use the Chinese Remainder Theorem and the prime factorization of
561.) ...
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