elementary-number-theory-practice-exams
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elementary-number-theory-practice-exams

Course Number: MATH 3307, Spring 2010

College/University: UT Arlington

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Math 3307 Practice Exams David Andrew Smith Abstract. Elementary number theory presents problems involving divisibility, prime numbers, Euclidean Algorithm, linear Diophantine equations, congruence problems, multiplicative functions, reciprocity laws, and various types of congruence applications. We also present several topics about elementary number theory including many reviews, practice exams, quizzes, and...

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3307 Math Practice Exams David Andrew Smith Abstract. Elementary number theory presents problems involving divisibility, prime numbers, Euclidean Algorithm, linear Diophantine equations, congruence problems, multiplicative functions, reciprocity laws, and various types of congruence applications. We also present several topics about elementary number theory including many reviews, practice exams, quizzes, and homework exercises. We briey describe some of the history of number theory in terms of some of the main mathematicians that contributed to the theory of numbers. Early contributors were Archimedes and the Pythagoreans (including Euclid), and then later on Fermat, Euler and Gauss. Number theory may very well be one of the oldest subjects of mathematics. Roughly, number theory is the mathematical treatment of questions related to the integers; that is, the numbers 0, 1, 1, 2, 2, 3, 3... and people has been manipulating them for thousands of years. The Ancient Greeks, in particular, Euclid (third century) and the Pythagoreans (sixth century B.C.) spent a considerable amount of attention to them; as well as Archimedes (third century). Indeed, one of the most important sets of numbers, the prime numbers; hold a key position in number theory since they are the building blocks of the integers; and perhaps the rst question that comes to mind is whether there are innitely many prime numbers. A proof of this amazing fact can be found in Euclids famous book: The Elements. Pierre de Fermat and Leonard Euler rekindled interest in number theory in the seventeenth and eighteenth centuries by using new (among others, calculus-related) techniques to arrive at important new results. Each new theorem of course, has led to many new questions and conjectures; and one of the fascinating aspects of number theory is that many unresolved questions can be understood with only a minor background in the subject. Even today there are many open problems; and some have a substantial reward for a solution! After Fermat and Euler, Carl Friedrich Gauss, one of the greatest mathematicians of all time, gave the rst modern treatment of number theory. He dened the notion of congruence, and distinguished its importance; in fact, its his notation and approach of number theory that we use today. Gausss many achievements in number theory are well documented; and its Gauss who coined the phrase number theory is the queen of the sciences. Interestingly enough, even in an elementary course of number theory other elds of mathematics come into play, such as the complex numbers, geometry, and abstract algebra. Various topics in elementary number theory including divisibility, congruence, quadratic reciprocity, and multiplicative functions. Upon completion of Number Theory students should be able to: (1) Prove statements and solve problems involving divisibility, prime numbers and the Euclidean Algorithm; (2) Solve linear Diophantine equations and various types of congruence problems, and use the theory of congruence in applications; (3) Apply properties of multiplicative functions such as the Euler phi-function and quadratic residues. 2010 Mathematics Subject Classication. Primary 00A07. Key words and phrases. elementary number theory, induction, divisibility, congruence, Fermats theorem, Eulers theorem, 1 quadratic reciprocity. Elementary Number Theory 1. Practice Exams 1, Spring 2007 (1) In using mathematical induction to prove n n(n+1) for all the statement: k=1 k = 2 positive integers n a correct step could be, (a) n k=1 (c) Yes, namely a = 6, b = 2, and c = 3. (d) Yes, namely a = 2, b = 6, and c = 3. (e) No. 1= 1(1+1) 2 n n(n+1) 2 (b) Assume k=1 k = positive integer n, ... (c) Assume k=1 k = positive integer n, ... (d) Assume k=1 k = tive integers n, ... (e) Assume k=1 k = tive integers n, ... n+1 n n+1 for some (5) Given the function f (n) = 2n2 + 29 with domain Z, which of the following is true: (a) f (2) is not prime (b) f (11) is not prime n(n+1) 2 for some n(n+1) 2 for all posi- (c) f (29) is prime (d) f (11) is prime and f (29) is not prime n(n+1) 2 for all posi(e) f (2) is prime and f (11) is not prime (6) Knowing that a = bq + r which of the following is true, (a) (a, b) = (b, q ) (b) (a, b) = (b, r) (c) (b, r) = (b, q ) (d) (a, q ) = (b, r) (e) none of the above (7) The integers 8a +3 and 5a +2 are relatively prime for all integers a because (a) (8a +3)x +(5a +2)y = 1 for all integers x and y. (b) (8a +3)x +(5a +2)y = 1 for all positive integers x and y. (c) (8a + 3)x + (5a + 2)y = 1 for some integers x and y with x + y = 3. (d) (8a + 3)x + (5a + 2)y = 1 for some integers x and y with x + y = 3. (e) (8a + 3)x + (5a + 2)y = 1 for some integers x and y with x + y = 1. (8) Given two nonzero integers a and b, (a, b) = d means (a) d|a and d|b (2) The Well-Ordering Principle states: (a) every set S of positive integers contains a least element. (b) every set S of integers contains a least element. (c) every nonempty set S of integers contains a least element. (d) every nonempty set S of positive integers contains a least element. (3) Given the two integers 128 and 38 the Division Algorithm yields unique integers q = 3 and r = 14 such that (a) 128 = 38 3 + 14 with 0 14 < 38 (b) 128 = 38 3 + 14 with 0 3 < 38 (c) 128 = 38 3 + 14 with 0 3 < 14 (d) 128 = 38 3 + 14 (e) 128 = 38 3 + 14 with 3 < 14 (4) Are there integers a, b, and c such that a|bc, but a b and a c? (a) Yes, namely a = 2, b = 10, and c = 10. (b) Yes, namely a = 10, b = 2, and c = 10. D. A. Smith Page 2 of 20 Elementary Number Theory (b) a|d and b|d (c) d|a, d|b, and if e is a common divisor of a and b then e|d. (d) d|a, d|b, and if e is a common divisor of a and b then d|e. (e) a|d, b|d, and if e is a common divisor of a and b then d|e. (9) If a and b are integers with a = 0, then a|b means (a) b = ac for all integers c. (b) b = ac for some integer c. (d) perform the Division Algorithm (e) nd (a, b) as a linear combination of a and b (11) The greatest common divisor of two nonzero integers a and b is characterized by (a) the smallest positive linear combination of a and b. (b) any linear combination of a and b. (c) the largest linear combination of a and b. (d) mathematical induction. (c) a = bc for some positive integer c. (e) none of the above. (d) a = bc for all integers c. (e) none of the above (10) The Sieve of Eratosthenes can be used to (a) perform the Euclidean Algorithm (b) determine (a, b) (c) list prime numbers (12) Let x be any real number greater than 1. Use mathematical induction to prove that (1 + x)n 1 + nx for all positive integers n. (13) Use the Euclidean Algorithm to nd (34709, 100313). (14) Show that if (a, b) = 1 and a|bc then a|c. D. A. Smith Page 3 of 20 Elementary Number Theory 2. Practice Exam 1, Spring 2008 (1) Using Mathematical Induction to prove, n 3 2 k=1 k = [n(n + 1)/2] for all positive integers n, a correct step would be: 3 2 (a) Assume k=1 k = [p(p + 1)/2] for all p, then ... 3 2 (b) Assume k=1 k = [p(p + 1)/2] for some p, then ... p+1 p (4) The product of three consecutive integers is divisible by 6 because (a) because a, (a + 1), and (a + 2) are all divisible by 6 where a is any integer (b) because one of a, (a + 1), and (a + 2) might be divisible by 6 where a is any integer (c) because one of a, (a 1), and (a + 1) is divisible by 2 and one is divisible by 3 where a is any integer (d) because a, (a + 1), and (a + 2) are all divisible by 2 and 3 where a is any integer (e) because a, (a + 1), and (a + 2) are all divisible by 2, 3, and 6 where a is any integer (5) For any integer a, (5a + 2, 7a + 3) = 1 because (a) (5(2) + 2, 7(2) + 3) = (12, 17) = 1 (b) 7(5a + 2) + 5(7a + 3) = 1 (c) not true for a = 71 (d) 7(5a + 2) 5(7a + 3) = 1 and so actually (5a + 2, 7a + 3) = 1. (e) not true for a = 5 (6) Apply the Euclidean Algorithm to nd (72, 258, 828). (a) 72 (c) 1 k=1 k 3 = [1(1 + 1)/2]2 p+1 (d) Assume k=1 k 3 = [(p + 1)(p + 2)/2]2 for all p, then ... (e) n k=1 k 3 = [n(n + 1)/2]2 (2) Using Mathematical Induction to prove, an + bn is divisible by a + b for all positive integers n, a correct step would be: (a) Assume a + b |an + bn for all n, then ... (b) Assume a + b ak+1 + bk+1 for some integer k, then ... (c) Assume a1 + b1 = a + b, then ... (d) Assume a + b ak + bk for some integer k, then ... (e) a1 + b1 = a + b (3) The product of two integers of the form 4k + 3 is of the form 4k + 1 because: (a) (4k + 3)(4k + 3) = 9 + 24k + 16k 2 = 4t + 1 for all integers t (b) (4k + 3)(4k + 3) = 9 + 24k + 16k 2 = 4t + 3 for some integer t (c) (4m + 3)(4n + 3) = 4(4nm + 3m + 3n + 2) + 1 for integers m and n (d) 9+24k +16k = 4t +1 for some integer t. (e) 9 + 24k + 16k 2 = 4t + 1 for all integers t. 2 (b) 9 (c) 12 (d) 2 (e) 6 (7) The power of 5 in the unique factorization of 11025 is (a) 2 D. A. Smith Page 4 of 20 Elementary Number Theory (b) 3 (c) 1 (d) 5 is not a factor (e) 4 (8) The primes in the unique factorization of 3460275 are (a) 3, 5, 7 (b) 3, 5, 13 (c) 3, 5, 7, 11 (d) 3, 5, 9, 11 (e) none of the above (9) Assume that a, b, and c are integers, which of the following properties is FALSE: (a) If a|b and b|c, then a|c. (b) If c|a and c|b, then c|(xa + yb) for any integers x and y. (c) If a|b and b|a, then a = b. (d) If c = 0 and a|b then ac|bc. (e) If a|b, then an |bn for any positive integer n. (10) Which of the following is NOT a Fibonacci Number: (a) 8 (b) 13 (c) 34 (d) 55 (e) 88 (11) Which of the following in NOT the same as a divides b: (a) a is a divisor of b (b) a is a factor of b (c) b is a multiple of a (d) (a, b) = 1 (e) there exists an integer k such that b = ak (12) Given the function f (n) = n2 n + 41 with domain Z, then which of the following is true: (a) f (11) is not prime (b) f (41) is prime (c) f (11) is prime and f (41) is not prime (d) f (41) is not prime and f (2) is not prime (e) f (2) is prime and f (11) is not prime (13) The Well-Ordering Principle states: (a) every set S of positive integers contains a least element. (b) every set S of integers contains a least element. (c) every nonempty set S of integers contains a least element. (d) every nonempty set S of positive integers contains a least element. (14) Given the two integers 128 and 38 the Division Algorithm yields unique integers q = 3 and r = 14 such that (a) 128 = 38 3 + 14 with 0 14 < 38 (b) 128 = 38 3 + 14 with 0 3 < 38 (c) 128 = 38 3 + 14 with 0 3 < 14 (d) 128 = 38 3 + 14 (e) 128 = 38 3 + 14 with 3 < 14 (15) Knowing that a = bq + r which of the following is true, (a) (a, b) = (b, q ) (b) (a, b) = (b, r) (c) (b, r) = (b, q ) D. A. Smith Page 5 of 20 Elementary Number Theory (d) (a, q ) = (b, r) (e) none of the above (16) Let fn be the Fibonnaci sequence, show that for all positive integers n, n j =1 f2j 1 = f2n . (17) Use the unique factorizations of n = 5248 and m = 1280 to determine the unique factorizations of (n, m)and [n, m]. (18) Use the Euclidean Algorithm to nd the greatest common divisor (105, 300) and then write this as a linear combination of these integers. D. A. Smith Page 6 of 20 Elementary Number Theory 3. Practice Exam 2, Spring 2007 (1) Given prime factorizations a = p1 e1 pn en and b = p1 f1 pn fn with possibly ei = 0 for some i and fi = 0 and some i, which of the following is true (a) [a, b] = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (b) (a, b) = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (c) [a, b] = p1 min(e1 ,f1 ) p2 min(e2 ,f2 ) pn min(en ,fn ) (d) (a, b) = p1 min(e1 ,f1 ) p2 min(e2 ,f2 ) pn min(en ,fn ) (e) [a, b]ab = (a, b) (2) The linear Diophantine equation ax + by = c has a solution if and only if (a) c|d, where d = (a, b) (b) d|a, where d = (a, c) (c) d|b, where d = (b, c) (d) d|c, where d = (a, b) (e) d|c, where c = (a, b) (3) Let n be a positive integer. Integers a and b are congruent modulo n means (a) a b is divisible by n and is denoted by a b(modn). (b) a b is divisible by n and is denoted by a b(mod(a, b)). (c) n is divisible by a b and is denoted by a b(modn). (d) n is divisible by a b and is denoted by a b n(modn). (e) a b is divisible by (a, b) and is denoted by a b(modn). (4) Let n > 1 be an integer. Then, which of the following is a FALSE statement: (a) If a c(modn) and b d(modn), then a b c d(modn) and ab cd(modn). (b) If a + c a + d(modn), then c d(modn). (c) If ac ad(modn) and (a, n) = 1, then c d(modn). (d) There exists an integer h such that ah 1(modn) if and only if (a, n) = 1. (e) If ac bc(modn), then a b(modn). (5) The least positive residue of 2200 modulo 47 is (a) 19 (b) 18 (c) 17 (d) 16 (e) 15 (6) Let x be an unknown in the linear congruence equation ax b(modn) and d = (a, n). Which of the following is a FALSE statement: (a) If d = 1, then there is precisely one (incongruent) solution. (b) If d b, then the congruence has no solution. (c) If d|b, then there are exactly d distinct (incongruent) solutions. (d) The congruence equation always has a solution. (e) There is exactly one (incongruent) solution when a and n are relatively prime. (7) Find the least positive residue of 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! modulo 7. (a) 1 D. A. Smith Page 7 of 20 Elementary Number Theory (b) 2 (c) 3 (a) there exists an integer k such that a = kb (b) a = kb, for all integers k (d) 4 (c) k = ab, for all integers k (e) 5 (d) (a, b) = k, for all integers k (8) A complete system of residues modulo 7 consisting entirely of even integers is (a) {0, 1, 2, 3, 4, 5, 6} (b) {0, 8, 16, 10, 18, 12, 6} (c) {0, 1, 16, 10, 18, 12, 6} (d) {0+7, 1+7, 2+7, 3+7, 4+7, 5+7, 6+7} (e) {0, 2, 4, 6, 8, 10, 12, 14} (9) The product of two integers of the form 6k + 5 is (a) never of the form 6k + 1 (b) sometimes of the form 6k + 1 (c) always of the form 6k + 1 (d) always of the form 6k + 2 (e) sometimes of the form 6k + 2 (10) If a|b, then (e) a b (11) Given the function f (n) = x2 x + 41 with domain Z, then which of the following is true: (a) f (41) is not prime (b) f (11) is not prime (c) f (41) is prime (d) f (11) is prime and f (41) is not prime (e) f (2) is prime and f (11) is not prime (12) Show that if a and b are positive integers, then (a, b) = (a + b, [a, b]). (13) Solve the linear Diophantine equation, 714x + 7007y = 7. (14) Solve the linear congruence equation 987x 610(mod1597). D. A. Smith Page 8 of 20 Elementary Number Theory 4. Practice Exam 2, Spring 2008 (1) Given prime factorizations a = p1 e1 pn en and b = p1 f1 pn fn with possibly ei = 0 and fi = 0 for some i, which of the following is true (a) [a, b] = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (b) (a, b) = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (c) [a, b] = p1 e1 +f1 p2 e2 +f2 pn en +fn (d) (a, b) = p1 e1 f1 p2 e2 f2 pn en fn (e) [a, b]ab = (a, b) (b) 1 (2) The linear Diophantine equation ax + by = c has a solution if and only if (a) c|d, where d = (a, b) (b) d|a, where d = (a, c) (c) d|b, where d = (b, c) (d) d|c, where d = (a, b) (e) d|c, where c = (a, b) (3) Let n be a positive integer. Integers a and b are congruent modulo n means (a) a b is divisible by n and is denoted by a b(modn). (b) a b is divisible by n and is denoted by a b(mod(a, b)). (c) n is divisible a by b and is denoted by a b(modn). (d) n is divisible by a b and is denoted by a b n(modn). (e) a b is divisible by (a, b) and is denoted by a b(modn). (4) Let n > 1 be an integer. Then, which of the following is a FALSE statement: (a) If a c(modn) and b d(modn), then a b c d(modn) and ab cd(modn). (c) 2 (d) 16 (e) 15 (6) Let x be an unknown in the linear congruence equation ax b(modn) and d = (a, n). Which of the following is a FALSE statement: (a) If d = 1, then there is precisely one (incongruent) solution. (b) If d b, then the congruence has no solution. (c) If d|b, then there are exactly d distinct (incongruent) solutions. (d) The congruence equation always has a solution. (e) There is exactly one (incongruent) solution when a and n are relatively prime. (7) Find the least positive residue of 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! modulo 7. (a) 1 (b) 2 (c) 3 (b) If a + c a + d(modn), then c d(modn). (c) If ac ad(modn) and (a, n) = 1, then c d(modn). (d) There exists an integer h such that ah 1(modn) if and only if (a, n) = 1. (e) If ac bc(modn), then a b(modn). (5) The least positive residue of 2204 modulo 17 is (a) 0 D. A. Smith Page 9 of 20 Elementary Number Theory (d) 4 (e) 5 (8) If x 5(mod6) then (a) x 2(mod3) (b) x 1(mod3) (c) x 0(mod3) (d) x 6(mod5) (e) x 0(mod2) (9) The product of two integers of the form 6k + 5 has the form (a) 6k + 1 (b) 6k + 2 (c) 6k + 3 (d) 6k + 4 (e) 6k + 5 (10) If a|b, then (a) there exists an integer k such that a = kb (b) a = kb, for all integers k (c) k = ab, for all integers k (d) (a, b) = k, for all integers k (e) a b (11) The system of linear congruences x a1 (modn1 ) , x a2 (modn2 ) , ...., x as (modns ) has a unique solution modulo N = n1 n2 ns provided (a) (a1 , a2 , ..., as ) = 1. (b) (n1 , n2 , ..., ns ) = 1. (c) a1 , a2 , ..., as are pairwise relatively prime. (d) n1 , n2 , ..., ns are pairwise relatively prime. (e) (ai , ni ) = 1 for all i with 1 i s. (12) If eggs are removed form a basket 2, 3, 4, 5, and 6, at a time, there remain, respectively, 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain. What is the least number of eggs that could have been in the basket? (13) Solve the linear Diophantine equation by either nding all solutions or by showing there are none for 17x + 13y = 100. (14) Solve the linear congruence equation 987x 610(mod1597). If you nd solutions explain why you have found them all. D. A. Smith Page 10 of 20 Elementary Number Theory 5. Practice Exam, Midterm Spring 2009 (1) Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps. (2) Show that the product of every two integers of the form 6k + 1 is also of the form 6k + 1. (3) Show there are innitely many primes of the form 6k + 5. (4) Show that j =1 (fj ) 2 = fn fn+1 whenever n is a positive integer. (5) Show that no integer of the form n3 + 1 is a prime, other than 2. n (6) Write out the unique prime factorization of 1494411775. Show each step. (7) Explain why a, a2 = a where a is a positive integer. Explain why 201 is not a prime. Explain why 11 is a prime number. (8) Solve the linear congruence 5x 7(mod57) using basic properties of congruence (no linear Diophantine equation). Show all your steps. (9) Solve the linear congruence 5x 15(mod35) by solving a linear Diophantine equation. Show all your steps. (10) Use the Chinese Remainder Theorem to solve the linear congruence equation 3x 11 (mod245). D. A. Smith Page 11 of 20 Elementary Number Theory 6. Practice Exam 3, Spring 2007 (1) Given prime factorizations a = p1 e1 pn en and b = p1 f1 pn fn with possibly ei = 0 for some i and fi = 0 and some i, which of the following is true (a) [a, b] = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (b) (a, b) = p1 max(e1 ,f1 ) p2 max(e2 ,f2 ) pn max(en ,fn ) (c) [a, b](a, b) = p1 min(e1 ,f1 ) p2 min(e2 ,f2 ) pn min(en ,fn ) (d) (a, b) = p1 min(e1 ,f1 ) p2 min(e2 ,f2 ) pn min(en ,fn ) (e) [a, b]ab = (a, b) (2) The linear Diophantine equation ax + by = c has a solution if and only if (a) c|d, where d = (a, b) (b) d|a, where d = (a, c) (c) d|b, where d = (b, c) (d) d|c, where d = (a, b) (e) d|c, where c = (a, b) (3) If eggs are removed from a basket 2, 3, 4, 5, or 6 at a time, there remain; respectively, 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain.. Find the least number of eggs that could have been in the basket. (a) 117 (b) 120 (c) 121 (d) 21 (e) none of the above (4) Let n > 1 be an integer. Then, which of the following is a FALSE statement: (c) every nonempty set S of integers contains a least element. (d) every nonempty set S of positive integers contains a least element. (6) Let x be an unknown in the linear congruence equation ax b(modn) and d = (a, n). Which of the following is a FALSE statement: (a) If d = 1, then there is precisely one (incongruent) solution. (b) If d b, then the congruence has no solution. (c) If d|b, then there are exactly d distinct (incongruent) solutions. (d) The congruence equation always has a solution. (e) There is exactly one (incongruent) solution when a and n are relatively prime. (7) Are there integers a, b, and c such that a|bc, but a b and a c? (a) If a c(modn) and b d(modn), then a b c d(modn) and ab cd(modn). (b) If a + c a + d(modn), then c d(modn). (c) If ac ad(modn) and (a, n) = 1, then c d(modn). (d) There exists an integer h such that ah 1(modn) if and only if (a, n) = 1. (e) If ac bc(modn), then a b(modn). (5) The Well-Ordering Principle states: (a) every set S of positive integers contains a least element. (b) every set S of integers contains a least element. D. A. Smith Page 12 of 20 Elementary Number Theory (a) Yes, namely a = 2, b = 10, and c = 10. (b) Yes, namely a = 10, b = 2, and c = 10. (c) Yes, namely a = 6, b = 2, and c = 3. (d) Yes, namely a = 2, b = 6, and c = 3. (e) No. (8) A complete system of residues modulo 6 consisting entirely of even integers is (a) {0, 1, 2, 3, 4, 5} (b) {0, 7, 8, 9, 10, 11} (c) {0, 1, 12, 14, 16} (d) {0 + 6, 1 + 6, 2 + 6, 3 + 6, 4 + 6, 5 + 6} (e) there does not exist one (9) Which of the following do you think is true: (a) if a is an inverse of a modulo m and b is an inverse of b modulo m, then a + b is an inverse of ab modulo m. (b) if a is an inverse of a modulo m and b is an inverse of b modulo m, then a b is an inverse of ab modulo m. (c) if a is an inverse of a modulo m and b is an inverse of b modulo m, then a is b an inverse of ab modulo n. (d) if a is an inverse of a modulo m and is an inverse of b modulo m, then b [a, b] = ab. (e) if a is an inverse of a modulo m and is an inverse of b modulo m, then b (a, b) = 1. (10) Determine the positive integers less than 14 that have an inverse modulo 14. (a) 1 (b) 1, 3, 5, (c) 1, 3, 5, 9, 11, 13 (d) 1, 3, 5, 9, 10, 11, 13 (e) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (11) The greatest common divisor of two nonzero integers a and b is characterized by (a) the smallest positive linear combination of a and b. (b) any linear combination of a and b. (c) the largest linear combination of a and b. (d) mathematical induction. (e) none of the above. D. A. Smith Page 13 of 20 Elementary Number Theory 7. Practice Exam, Final Exam Spring 2007 (1) If a, b and c = 0 are integers then a|b if and only if ac|bc is (a) always true. (b) true if a, b and c are positive. (c) true if a, b and c are negative. (d) true only if c = (a, b). (e) always false. (2) Given the function f (n) = n2 n + 41 with domain Z, then which of the following is true: (a) f (11) is not prime (b) f (41) is prime (c) f (11) is prime and f (41) is not prime (d) f (41) is not primeand f (2) is not prime (e) f (2) is prime and f (11) is not prime (3) What is the greatest common divisor of a and a2 + 2? (a) 1 or 2 (b) a or a2 (c) 1 or a (d) a or 2 (e) only 1 (4) Given two nonzero integers a and b then (a, b) = c means (a) c|a and c|b (b) a|c and b|c (c) c|a, c|b, and if e is a common divisor of a and b then e|c. (d) c|a, c|b, and if e is a common divisor of a and b then c|e. 2 (e) a|c, b|c, and if e is a common divisor of a and b then c|e. (5) Every integer can be written as the product of possibly a square and a square-free integer is a consequence of which theorem: (A square-free integer is an integer that is not divisible by any perfect square other than 1). (a) Euclidean Algorithm (b) Innitude of Primes (c) Fermats Theorem (d) Fundamental Theorem of Arithmetic (e) Eulers Theorem (6) To solve the linear Diophantine equation 12x + 18y = 50 we (a) perform the Euclidean Algorithm to nd x and y such that 12x + 18y = 50. (b) determine there are no solutions because (12, 18) 50. (c) determine there are 6 incongruent solutions since (12, 18) = 6. (d) perform the Euclidean Algorithm to nd x and y such that 12x + 18y = 2 and then multiply by 25. (e) determine there are 50 incongruent solutions and we use the Chinese Remainder theorem to nd them (7) For which positive integers m is the congruence equation 27 5(modm) true? (a) 22 (b) 1, 2, 11, and22 (c) 1, 2, 11, and 22 (d) 5 (e) 1 and 5 D. A. Smith Page 14 of 20 Elementary Number Theory (8) For which integers c with 0 c < 30 is 12x c(mod30) solvable? (a) c = 0, 6, 12, 18, 24 (b) c = 0, 6, 12, 18 (c) c = 6, 12 (d) c = 0, 6, 12, 15, 18, 24 (e) c = 0, 6, 12, 15, 18 (9) When nding an integer that leaves a remainder of 1 when divided by either 2 or 5, but that is divisible by 3 we use (a) the Euclidean algorithm to show no such integer exists (b) the Euclidean algorithm to show such an integer exists (c) the Chinese Remainder to show no such integer exists (d) the Chinese Remainder to show such an integer exists (e) the Fundamental Theorem Arithmetic to factor 21. (10) Suppose x0 is a solution to the polynomial congruence f (x) 0 (modpn ) . In an attempt to lift x0 to a solution for f (x) 0 modpn+1 we (a) always guess 1, 2, ...., pn+1 and check for (c) 7 (d) 8 (e) 11 (12) On April 13, 2029 it is known that the asteroid 2004 MN4 will have a 2.2% chance of hitting planet Earth. Determine the day of the week. (a) Monday (b) Tuesday (c) Wednesday (d) Thursday (e) Friday (13) Find the least positive residue of 8 9 10 11 12 13 modulo 7. (a) 1 (b) 3 (c) 4 (d) 5 (e) 6 (14) Find the last digit in the decimal expansion of 7999,999 . (a) 1 (b) 3 (c) 7 (d) 8 (e) 9 (15) Which of the following statements is FALSE: (a) (n) is even provided n is a positive integer (b) (mn) = (m)(n) provided m and n are positive integers (b) nd the inverse for f (x0 ) working modpn . (c) factor each of the coecients of f (d) compute the gcd of the coecients of f (e) compute the derivative of f (11) The highest power of 2 that divides 4345844657033088 is (a) 13 (b) 5 D. A. Smith Page 15 of 20 Elementary Number Theory (c) (pn ) = pn pn1 provided p is a prime and n is a positive integer (d) (p) = p 1 provided p is a prime integer (e) mk = mk1 (m) provided m and k are positive integers (16) Prove n < 2n whenever n is a positive integer. (17) Solve the congruence 3x 5(mod16) by writing a linear Diophantine equation and solving it. (18) Solve the congruence 5x 3(mod14) by using Eulers theorem. (19) What is the remainder when 18! is divided by 437? (20) Show that (5186) = (5187) = (5188). D. A. Smith Page 16 of 20 Elementary Number Theory 8. Practice Exam, Final Exam Spring 2008 (1) The solutions to the linear Diophantine equation 25x 12y = 15 are (a) x = 15 12t, y = 30 25t where t is an integer (b) x = 15 + 12t, y = 30 25t where t is an integer (c) x = 15 12t, y = 30 + 25t where t is an integer (d) x = 5 12t, y = 15 25t where t is an integer (e) x = 5 + 12t, y = 15 25t where t is an integer (2) Find the last digit in the decimal expansion of 3207 + 3. (a) 0 (b) 1 (c) 3 (d) 5 (e) 7 (3) Which of the following satises (n) = 96? (a) 194 and 196 (b) 2002 (c) 200 (d) 194 only (e) 194 and 195 (4) Determine the least positive residue for 3298 + 523 modulo 17. (a) 0 (b) 1 (c) 2 (d) 3 (8) Determine the day of the week for December 28th, 1996. 2 (e) 4 (5) Which of the following is true: (a) 7 19 (b) 7 19 2 5 = =1 19 7 = 5 7 = 7 5 = (c) 7 19 2 5 = 179 = 1 = = 1 19 7 = 5 7 = 7 5 = = (d) 7 19 2 5 5 7 = 7 5 = (e) 7 19 2 5 2 5 = 179 = 1 19 7 = 5 7 = 7 5 = = = 1 = 5 7 = 7 5 = (6) Given x2 a(modp) with x a(modp) (a) has only one solution a p = 1 then (b) has exactly two solutions (c) has no solution (d) has two solutions, if p 1(mod4). (e) has two solutions, if p 3(mod4). (7) Which of the following statements is FALSE: (a) if n|m then (n)|(m) (b) (mn) = (m)(n) provided m and n are positive integers (c) (pn ) = pn pn1 provided p is a prime and n is a positive integer (d) (p) = p 1 provided p is a prime integer (e) mk = mk1 (m) provided m and k are positive integers D. A. Smith Page 17 of 20 Elementary Number Theory (a) Monday (b) Tuesday (c) Friday (d) Saturday (e) Sunday (9) Prove that if a b(modm) then ak bk (modm) for every positive integer k. (10) Use the Euclidean algorithm to nd (1372, 490) and write the GCD as a linear combination of 1372 and 490. (11) Solve the polynomial congruence equation 2x2 3x + 12 0(mod343). (12) Solve the system 2x 3(mod4), 5x 6(mod7), 9x 10(mod11) using the Chinese Remainder theorem. D. A. Smith Page 18 of 20 Elementary Number Theory 9. Practice Exam, Final Exam Spring 2009 (1) Use mathematical induction to show that n k for every positive integer n, = i=1 2 n+1 2 2. (2) Given nonzero integers a, b, and c show that a|b and a|c implies a|(bx + cy ) for any integers x and y. (3) Use the Eucidean algorithm to nd d = (500, 50, 40) and write d as a linear combination of the three given integers. (4) Explain why the linear Diophatine equation 2x 101y = 82 is solvable or not solvable. If possible nd all solutions. (5) Explain why the linear congruence equation 3x 81(mod910) is solvable or not solvable. If possible solve it. (6) Construct the multiplication and addition tables for modular arithmetic for n = 6. (7) Using the Chinese Remainder Theorem, solve the system of linear congruence equations 2x 3(mod4), 3x 5(mod6), and 4x 1(mod7). (8) Find the least positive residue of 9! + 10! + 11! + 12! + 13! modulo 11. (9) Find the second to last digit in the decimal expansion of 4352 . (10) Use Eulers theorem to nd any x that satises the linear congruence equation 3x 13(mod17). (11) Determine (10440125). (12) Find all the quadratic residues of 13. (13) Determine if x2 105(mod1009) is solvable using Legendre symbols. (14) Use Hensels lifting theorem and the Chinese Remainder Theorem to solve the quadratic congruence equation 3x2 + 11x + 2 0(mod72). D. A. Smith Page 19 of 20 Elementary Number Theory References [1] Rosen, Kenneth (2005) Elmentary Number Theory and its Applications (5th Edition). Pearson/Addison Wesley. ISBN 978-0321237071. Current address : University of Texas at Arlington, Mathematics Department, 411 South Nedderman Drive, Arlington, Texas 76019, 442 Pickard Hall E-mail address : davidsmith@uta.edu URL: http://www.uta.edu/faculty/dsmith D. A. Smith Page 20 of 20

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UT Arlington - MATH - 3301
A FOUNDATIONS OF GEOMETRY STUDY GUIDEDAVID ANDREW SMITHAbstract. An introduction to the foundations of geometry is an undergraduate course at many universities. Topics included in such a course can vary widely but a common approach is to study the basic
UT Arlington - MATH - 1426
Math 1426 Final Exam - Version 1 Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _Fall 2007Check the appropriate section: _ _ _ 018 Mr. Smith, MWF 8am 024 Mr. Smith, MW 1pm 030 Mr. Martines _ _ _ 021
UT Arlington - MATH - 1426
Math 1426 Midterm 1 Version 1 Print your name legibly as it appears on the class rolls:Fall 2007Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _Check the appropriate section: 018 Mr. Smith, MWF 8am 021 Dr. Shan 024 Mr. Smith, MW 1pm 027 Dr. Epperson 030 M
UT Arlington - MATH - 1426
Math 1426 Midterm 2 Version 1 Print your name legibly as it appears on the class rolls:Fall 2007Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _Check the appropriate section: 018 Mr. Smith, MWF 8am 021 Dr. Shan 024 Mr. Smith, MW 1pm 027 Dr. Epperson 030 M
UT Arlington - MATH - 1426
Math 1426Final Exam Version AFall 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Krueger 004 Dr. Shan 007 Dr. Jorgensen 010 Dr. Epperson 013 Mr. Clanton
UT Arlington - MATH - 1426
Math 1426Midterm 1 Version AFall 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Krueger 004 Dr. Shan 007 Dr. Jorgensen 010 Dr. Epperson 013 Mr. Clanton
UT Arlington - MATH - 1426
Math 1426Midterm 2 Version AFall 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Krueger 004 Dr. Shan 007 Dr. Jorgensen 010 Dr. Epperson 013 Mr. Clanton
UT Arlington - MATH - 1426
Math 1426Midterm 1AFall 2009Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Pankavich 002 Mr. Smith MWF 8 am 003 Dr. Krueger 004 Mr. Smith MW 1 pm 005 Dr. Ki
UT Arlington - MATH - 1426
Math 1426Midterm 2 Version AFall 2009Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 002 Mr. Smith MWF 8 am 003 Dr. Krueger 004 Mr. Smith MW 1 pm 005 Dr. Kirby 006 D
UT Arlington - MATH - 1426
Math 1426Final Exam Version ASpring 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _ Check the appropriate section: 001 Dr. Shan 004 Dr. Krueger 007 Mr. Smith 010 Mr. Martines 013 Dr. Lin*Fill
UT Arlington - MATH - 1426
Math 1426Midterm 1 - Version ASpring 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _ Check the appropriate section: 001 Dr. Shan 004 Dr. Krueger 007 Mr. Smith 010 Mr. Martines 013 Dr. Lin *WRI
UT Arlington - MATH - 1426
Math 1426Midterm 2 Version ASpring 2008Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: _ _ _ _ _ _ _ _ _ _ Check the appropriate section: 001 Dr. Shan 004 Dr. Krueger 007 Mr. Smith 010 Mr. Martines 013 Dr. LinFill i
UT Arlington - MATH - 1426
Math 1426Midterm 1 Version ASpring 2009Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Shan 004 Dr. Krueger 007 Dr. Jha 013 Dr. VancliffFill in your scantro
UT Arlington - MATH - 1426
Math 1426Midterm 2 Version ASpring 2009Print your name legibly as it appears on the class rolls: Last _ First _ ID Number: 1 0 0 0 _ _ _ _ _ _ Check the appropriate section: 001 Dr. Shan 004 Dr. Krueger 007 Dr. Jha 013 Dr. VancliffFill in your scantro
UT Arlington - BIOL - 3444
Biology 3444-001 General Microbiology Fall 2008 General information: Instructor: Dr. Thomas Chrzanowski Office: 243 Life Sciences Phone: 272.2404 Email: Chrz@uta.edu Office Hours: Monday and Wednesday 8:00-9:00 am Textbook: Brock Biology of Microorganisms
UT Arlington - BIOL - 3339
Evolution (Biol 3339/5311) - Chippindale Lecture notes, Spring 2010Intro, review syllabus, etc. Evolution: process that explains diversity of life on Earth - all life on Earth descends from a common ancestor. Precise definition of evolution: Change in al
UT Arlington - MATH - 1325
Math Review 1 Geometry Here are some areas of standard geometric gures that you should remember: 2. Rectangle A = base height 3. Circle A = radius2Triangle Rectangle Circle radius diameter height base1. Triangle A = 1 base height 2height base The vol
UT Arlington - PHYS - 1441
Unit 1 Uncertainties There are always uncertainties involved in measurements and in calculations that involve measurements with uncertainties. There are also methods that can help reduce uncertainties of measure quantities. One of these methods is to make
UT Arlington - PHYS - 1441
Unit 3 Help FileTopics 1) How to use graphical methods to determine an equilibrant. 2) How to use analytical methods to determine an equilibrant. i.e mathematically 3) Solving for two unknown vectors using simultaneous equations. 4) Addition notes for th
UT Arlington - PHYS - 1402
Atomic Spectra Lab 9In the early 20th century, gaseous elements were found to have a unique atomic spectra when subjected to a high voltage so that they emitted a light. This light when looked at through a prism or a transmission grating shows that the l
UT Arlington - PHYS - 1402
Electric FieldsTheory: An electric field is a region in space in which electric forces act on electric charges, if present. The electric field strength for any point in space is defined as the net electric (Coulomb) force per unit of positive charge acti
UT Arlington - PHYS - 1402
Ohms LawIn a conductor where a electric field is established electrons are free to move. The electrons move from a place where their electrical potential is high to a place where they have less electrical potential, the electrons are said to flow at this
UT Arlington - PHYS - 1402
Magnetic FieldsBasic principles of magnetism are that like poles repel; unlike poles attract. Also a magnet produces a magnetic field which surrounds and permeates the area around the magnet. This field can penetrate through non-magnetic material such as
UT Arlington - PHYS - 1402
Reflection and RefractionTheory: Whenever a wave traveling in some medium encounters an interface or boundary with another medium either (or both) of the processes of (1) reflection and (2) refraction may occur if the speed of the wave is different in th
UT Arlington - PHYS - 1401
Lab 1Metric system and unit converionsThis lab is intended as an introduction to a new measurement system, scientific notation, units, and conversions. Hopefully upon completion of this lab exercise the foundation of your knowledge of what might be some
UT Arlington - PHYS - 1401
Lab 2 - Motion in One dimensionIn this lab, you will explore motion in one dimension. A computer will aid in acquiring the data and displaying the data to be analyzed. Theory: If an object is moving at a constant acceleration over a fixed period of time,
UT Arlington - PHYS - 1401
The Second Law of MotionTheory: The Second Law of Motion states that: The acceleration of an object is directly proportional to the net force applied to the object and inversely proportional to the mass of the object. If the acceleration is in units of m
UT Arlington - PHYS - 1401
Studying the Laws of Conservation of Momentum and Energy using a Ballistic PendulumTheory: Conservation of Momentum: Momentum is the product of an objects mass and direction. Momentum is also a vector therefore the direction of the object is important to
UT Arlington - PHYS - 1401
Projectile Motion In this experiment we will study motion in two-dimensions. An object which has motion in both the X and Y direction has a two dimensional motion. We will first determine at what velocity the ball is being fired from the firing mechanism,
UT Arlington - PHYS - 1401
Archimedes Principle and Applications Objectives: Upon successful completion of this exercise you will have . 1. . utilized Archimedes principle to determine the density and specific gravity of a variety of substances. 2. . utilized Archimedes principle t
CSU Northridge - BIO - 310
Cell theory Smallest living thing Living things are composed of cells Gameteseggs and sperm in gonadsovaries or testesMeiosis Mitosiscloning,cells, recombination 200 types of somatic cells rbc has no nucleus, no need for nucleus some cells have more nucl
University of Texas - CH - 302
raymis (amr2936) H05: Equilibrium 2 Mccord (52445) This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 10.0 points A 2.000 liter vessel is lled with 4.000 moles of
University of Toronto - CHM138 - CHM138
Academy of Art University - CULINARY - 43435325
CULINARYChicken ConsommeRecipe 12.3 Mise en Place: Peel and chop onions, carrots and celery for mirepoix. Seed and dice tomato. Prepare onion brulee and herb sachet. Yields: 4 qt ( 4 l )Ingredients Egg whites Ground chicken Mirepoix Tomatoes, seeded an
Limestone - CODING - 456
Thse de DoctoratL'COLE POLYTECHNIQUEpour obtenir le titre de prsente DOCTEUR EN SCIENCESSpcialit Informatiquesoutenue le 28 novembre 2006 parMathieu CLUZEAUReconnaissance d'un schma de codageJury RapporteursThierry BERGER Vladimir SIDORENKO Unive
Berkeley - MSE - 103
mse 103Phase Transformations and Kinetics University of CaliforniaSpring 2009Problem set 1 Solution Due: Wednesday, January 28th, by 11:10am before lecture begins; late homework will not be accepted 1. The schematic below shows a hypothetical P-T diagr
Nottingham Trent - EEE - deeen
ASSIGNMENT 1EEE 2102Make a self-research on Full Wave R-L load Rectifier and use your knowledge to solve the following simple and short problem. Due by first Thursday after holiday.No excuses accepted!Given a source of Vm = 100V, 50Hz, R = 10 and L =
Johns Hopkins - WRITING - 220.200
OneIN THE beginning was the voice of Father.&quot;Emaleth!&quot; whispering close to her mother's belly while hermother slept. And then singing to her, the long songs of the past. Songsof the Glen of Donnelaith and of the castle, and of where they wouldsometim
Johns Hopkins - WRITING - 220.200
I'M THE VAMPIRE LESTAT. REMEMBER ME? THE vampire who became a super rock star, the one who wrote the autobiography? The one with the blond hair and the gray eyes, and the insatiable desire for visibility and fame? You remember. I wanted to be a symbol of
Johns Hopkins - WRITING - 220.200
Johns Hopkins - WRITING - 220.200
Harry Potter and the Sorcerers Stone By J.K. RowlingCHAPTER ONEThe Boy Who Lived Mr. and Mrs. Dursley, of number four, Privet Drive, were proud to say that they were perfectly normal, thank you very much. They were the last people youd expect to be invo
Johns Hopkins - WRITING - 220.200
Harry Potter and the Chamber of Secrets By J.K. RowlingCHAPTER ONEThe Worst Birthday Not for the first time, an argument had broken out over breakfast at number four, Privet Drive. Mr. Vernon Dursley had been woken in the early hours of the morning by a
Johns Hopkins - WRITING - 220.200
Harry Potter and the Prisoner of Azkaban By J.K. RowlingCHAPTER ONEOwl Post Harry Potter was a highly unusual boy in many ways. For one thing, he hated the summer holidays more than any other time of year. For another, he really wanted to do his homewor
Johns Hopkins - WRITING - 220.200
Harry Potter and the Goblet of Fire By J.K. RowlingCHAPTER ONEThe Riddle House The villagers of Little Hangleton still called it the Riddle House, even though it had been many years since the Riddle family had lived there. It stood on a hill overlooking
Johns Hopkins - WRITING - 220.200
Harry Potter and the Order of the Phoenix By J.K. RowlingCHAPTER ONEDudley Demented The hottest day of the summer so far was drawing to a close and a drowsy silence lay over the large, square houses of Privet Drive. Cars that were usually gleaming stood
Johns Hopkins - WRITING - 220.200
Harry Potter and the Half-Blood Prince By J.K. RowlingCHAPTER ONEThe Other Minister It was nearing midnight and the Prime Minister was sitting alone in his office, reading a long memo that was slipping through his brain without leaving the slightest tra
Johns Hopkins - WRITING - 220.200
RESURRECTIONby A. E. van VogtTHE GREAT ship poised a quarter of a mile above one of the cities. Below was a cosmic desolation. As he floated down in his energy bubble, Enash saw that the buildings were crumbling with age. &quot;No signs of war damage!&quot; &quot;The
Johns Hopkins - WRITING - 220.200
TO FORD McCORMACK I ON AND ON Coeurl prowled. The black, moonless, almost starless night yielded reluctantly before a grim reddish dawn that crept up from his left. It was a vague light that gave no sense of approaching warmth. It slowly revealed a nightm
Johns Hopkins - WRITING - 220.200
A.E. Van Vogt THE RAT AND THE SNAKE Mark Gray's main pleasure in life was feeding rats to his pet python.He kept the python in a blocked-off room in the old house in which helived alone. Each mealtime, he would put the rat in a narrow tunnel he hadr
Johns Hopkins - WRITING - 220.200
ASYLUMbyA. E. VAN VOCTI INDECISION WAS dark in the mans thoughts as he walked across the spaceship control room to the cot where the woman lay so taut and so still. He bent over her; he said in his deep voice: Were slowing down, Merla. No answer, no mo
Johns Hopkins - WRITING - 220.200
A.E. Van Vogt THE BARBARIAN In his initial address to the Patronate, following his return fromVenus, Tews said among other things, &quot;It is difficult for us to realze,but Linn is now without formidable enemies anywhere. Our opponents on Marsand Venus
Johns Hopkins - WRITING - 220.200
BLACK DESTROYERbyA. E. VAN VOGTON AND ON COEURL PROWLED! The black, moonless, almost starless night yielded reluctantly before a grim reddish dawn that crept up from his left. A vague, dull light it was, that gave no sense of approaching warmth, no com
Johns Hopkins - WRITING - 220.200
CONCEALMENT By A E van Vogt The Earth ship came so swiftly around the planetless Gisser sun that the alarm system in the meteorite weather station had no time to react. The great machine was already visible when Watcher grew aware of it. Alarms must have
Johns Hopkins - WRITING - 220.200
A.E. Van Vogt ERSATZ ETERNAL Grayson removed the irons from the other's wrists and legs. &quot;Hart!&quot;he said sharply. The young man on the cot did not stir. Grayson hesitated and thendeliberately kicked the man. &quot;Damn you, Hart, listen to me! I'm releas
Johns Hopkins - WRITING - 220.200
THE WEAPONS SHOP by A. E. VAN VOGTTHE VILLAGE at night made a curiously timeless picture. Fara walked contentedly beside his wife along the street. The air was like wine; and he was thinking dimly of the artist who had come up from Imperial City, and mad
Johns Hopkins - WRITING - 220.200
VERSION 1.0 dtd 033000ISAAC ASIMOVThe Bicentennial ManIn the introduction to this Nebula Awards volume it was mentioned that science fiction writers-successful science fiction writers-are unique. No one, however, is quite as unusual as Isaac Asimov. He
Johns Hopkins - WRITING - 220.200
Dune: Nighttime Shadows on Open Sandby Brian Herbert and Kevin J. Anderson Nature commits no errors; right and wrong are human categories. Pardot Kynes, Arrakis LecturesMonotonous days. The three-man Harkonnen patrol cruised over the golden swells of
Johns Hopkins - WRITING - 220.200
Dune:House HarkonnenBrian Herbert and Kevin J. Anderson October 2000To our mutual friend Ed Kramer, without whom this project would never have come to fruition. He provided the spark that brought us together.ACKNOWLEDGMENTSJan Herbert, with appreciat
Johns Hopkins - WRITING - 220.200
Dune:House AtreidesBrian Herbert and Kevin J. Anderson October 1999This book is for our mentor, Frank Herbert, who was every bit as fascinating and complex as the marvelous Dune universe he created.ACKNOWLEDGMENTSEd Kramer, for being the bridge that
Johns Hopkins - WRITING - 220.200
Dune: Nighttime Shadows on Open Sandby Brian Herbert and Kevin J. Anderson 1Nature commits no errors; right and wrong are human categories. -Pardot Kynes, Arrakis LecturesMonotonous days. The three-man Harkonnen patrol cruised over the golden swel
Johns Hopkins - WRITING - 220.200
Dune: House CorrinoBrian Herbert &amp; Kevin J AndersonThe axis of spin for the planet Arrakis is at right angles to the radius of its orbit. The world itself is not a globe, but more a spinning top somewhat fat at the equator and concave toward the poles.