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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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