DISCRETE MATHEMATICS WITH APPLICATION (
5th Edition
ISBN: 9780357097717
Author: EPP
Publisher: CENGAGE L
expand_more
expand_more
format_list_bulleted
Question
Chapter 8.4, Problem 4TY
To determine
To fill in the blanks of the statement “If a , n and k are positive integers with n>1, an efficient way to compute ak(mod n) is to write k as a ________ and use the facts about computing products and powers modulo n .”
Expert Solution & Answer

Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
DO these math problems without ai, show the solutions as well. and how you solved it. and could you do it with in the time spand
The Cartesian coordinates of a point are given.
(a) (-8, 8)
(i) Find polar coordinates (r, 0) of the point, where r > 0 and 0 ≤ 0 0 and 0 ≤ 0 < 2π.
(1, 0) =
(r.
= ([
(ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 ≤ 0 < 2π.
(5, 6) =
=([
The Cartesian coordinates of a point are given.
(a) (4,-4)
(i) Find polar coordinates (r, e) of the point, where r > 0 and 0 0 and 0 < 0 < 2π.
(r, 6) =
X
7
(ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 0 < 2π.
(r, 0) =
X
Chapter 8 Solutions
DISCRETE MATHEMATICS WITH APPLICATION (
Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TYCh. 8.1 - Prob. 3TYCh. 8.1 - Prob. 4TYCh. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES
Ch. 8.1 - Let X={a,b,c}. Define a relation J on P(X) as...Ch. 8.1 - Define a relation R on Z as follows: For all...Ch. 8.1 - Prob. 8ESCh. 8.1 - Let A be the set of all strings of 0’s, 1’s, and...Ch. 8.1 - Let A={3,4,5} and B={4,5,6} and let R be the “less...Ch. 8.1 - Let A={3,4,5} and B={4,5,6} and let S be the...Ch. 8.1 - Prob. 12ESCh. 8.1 - Prob. 13ESCh. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Prob. 16ESCh. 8.1 - Prob. 17ESCh. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Exercises 19-20 refer to unions and intersections...Ch. 8.1 - Prob. 20ESCh. 8.1 - Define relation R and S on R as follows:...Ch. 8.1 - Prob. 22ESCh. 8.1 - Prob. 23ESCh. 8.1 - Prob. 24ESCh. 8.2 - For a relation R on a set A to be reflexive means...Ch. 8.2 - For a relation R on a set A to be symmetric means...Ch. 8.2 - For a relation R on a set A to be transitive means...Ch. 8.2 - Prob. 4TYCh. 8.2 - Prob. 5TYCh. 8.2 - Prob. 6TYCh. 8.2 - Prob. 7TYCh. 8.2 - Prob. 8TYCh. 8.2 - Prob. 9TYCh. 8.2 - Prob. 10TYCh. 8.2 - Prob. 1ESCh. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - Prob. 3ESCh. 8.2 - Prob. 4ESCh. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9—33, determine whether the given relation is...Ch. 8.2 - In 9—33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 15ESCh. 8.2 - Prob. 16ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 18ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 20ESCh. 8.2 - Prob. 21ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 24ESCh. 8.2 - In 9-33, determine whether the given is reflexive...Ch. 8.2 - Prob. 26ESCh. 8.2 - Prob. 27ESCh. 8.2 - Prob. 28ESCh. 8.2 - Prob. 29ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 31ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 34-36, assume that R is a relation on a et A....Ch. 8.2 - Prob. 35ESCh. 8.2 - Prob. 36ESCh. 8.2 - Prob. 37ESCh. 8.2 - Prob. 38ESCh. 8.2 - Prob. 39ESCh. 8.2 - Prob. 40ESCh. 8.2 - Prob. 41ESCh. 8.2 - In 37-42, assume that R and S are relations on a...Ch. 8.2 - In 43-50, the following definitions are used: A...Ch. 8.2 - Prob. 44ESCh. 8.2 - Prob. 45ESCh. 8.2 - Prob. 46ESCh. 8.2 - Prob. 47ESCh. 8.2 - In 43-50, the following definitions are used: A...Ch. 8.2 - Prob. 49ESCh. 8.2 - Prob. 50ESCh. 8.2 - Prob. 51ESCh. 8.2 - In 51—53, R, S, and T are relations defined on...Ch. 8.2 - Prob. 53ESCh. 8.2 - Prob. 54ESCh. 8.2 - Prob. 55ESCh. 8.2 - Prob. 56ESCh. 8.3 - For a relation on a set to be an equivalence...Ch. 8.3 - The notation m=n(modd) is...Ch. 8.3 - Prob. 3TYCh. 8.3 - Prob. 4TYCh. 8.3 - Prob. 5TYCh. 8.3 - Prob. 6TYCh. 8.3 - Prob. 1ESCh. 8.3 - Prob. 2ESCh. 8.3 - Prob. 3ESCh. 8.3 - In each of 3—6, the relation R is an equivalence...Ch. 8.3 - Prob. 5ESCh. 8.3 - In each of 3-6, the relation R is an equivalence...Ch. 8.3 - Prob. 7ESCh. 8.3 - Prob. 8ESCh. 8.3 - Prob. 9ESCh. 8.3 - In each of 7-14, relation R is an equivalence...Ch. 8.3 - Prob. 11ESCh. 8.3 - In each of 7-14, relation R is an equivalence...Ch. 8.3 - In each of 7-14, the relation R is an equivalence...Ch. 8.3 - In each of 7—14, the relation R is an equivalence...Ch. 8.3 - Determine which of the following congruence...Ch. 8.3 - Let R be the relation of congruence modulo 3....Ch. 8.3 - Prob. 17ESCh. 8.3 - Prob. 18ESCh. 8.3 - In 19-31, (1) prove that the relation is an...Ch. 8.3 - Prob. 20ESCh. 8.3 - Prob. 21ESCh. 8.3 - Prob. 22ESCh. 8.3 - Prob. 23ESCh. 8.3 - In 19-31. (1) prove that the relation is an...Ch. 8.3 - In 19-31,(1) prove that the relation is an...Ch. 8.3 - Prob. 26ESCh. 8.3 - Prob. 27ESCh. 8.3 - Prob. 28ESCh. 8.3 - Prob. 29ESCh. 8.3 - Prob. 30ESCh. 8.3 - In 19—31, (1) prove that the relation is an...Ch. 8.3 - Prob. 32ESCh. 8.3 - Prob. 33ESCh. 8.3 - Prob. 34ESCh. 8.3 - Prob. 35ESCh. 8.3 - Prob. 36ESCh. 8.3 - Prob. 37ESCh. 8.3 - Prob. 38ESCh. 8.3 - Prob. 39ESCh. 8.3 - Prob. 40ESCh. 8.3 - Prob. 41ESCh. 8.3 - Prob. 42ESCh. 8.3 - Prob. 43ESCh. 8.3 - Let A=Z+Z+ . Define a relation R on A as follows:...Ch. 8.3 - Prob. 45ESCh. 8.3 - Let R be a relation on a set A and suppose R is...Ch. 8.3 - Refer to the quote at the beginning of this...Ch. 8.4 - When letters of the alphabet are encrypted using...Ch. 8.4 - Prob. 2TYCh. 8.4 - Prob. 3TYCh. 8.4 - Prob. 4TYCh. 8.4 - Prob. 5TYCh. 8.4 - Prob. 6TYCh. 8.4 - Prob. 7TYCh. 8.4 - Prob. 8TYCh. 8.4 - Fermat’s little theorem says that if p is any...Ch. 8.4 - Prob. 10TYCh. 8.4 - Prob. 1ESCh. 8.4 - Use the Caesar cipher to encrypt the message AN...Ch. 8.4 - Prob. 3ESCh. 8.4 - Let a=68, b=33, and n=7. Verify that 7|(68-33)....Ch. 8.4 - Prove the transitivity of modular congruence. That...Ch. 8.4 - Prob. 6ESCh. 8.4 - Verify the following statements. 128=2(mod7) and...Ch. 8.4 - Verify the following statements. 45=3 (mod 6) and...Ch. 8.4 - Prob. 9ESCh. 8.4 - In 9-11, prove each of the given statements,...Ch. 8.4 - In 9-11, prove each of the given statements,...Ch. 8.4 - Prove that for every integer n0,10n=1(mod9) . Use...Ch. 8.4 - Prob. 13ESCh. 8.4 - Prob. 14ESCh. 8.4 - Prob. 15ESCh. 8.4 - In 16-18, use the techniques of Example 8.4.4 and...Ch. 8.4 - Prob. 17ESCh. 8.4 - Prob. 18ESCh. 8.4 - Prob. 19ESCh. 8.4 - Prob. 20ESCh. 8.4 - Prob. 21ESCh. 8.4 - In 19-24, use the RSA cipher from Examples 8.4.9...Ch. 8.4 - Prob. 23ESCh. 8.4 - Prob. 24ESCh. 8.4 - Prob. 25ESCh. 8.4 - Prob. 26ESCh. 8.4 - In 26 and 27, use the extended Euclidean algorithm...Ch. 8.4 - Prob. 28ESCh. 8.4 - Prob. 29ESCh. 8.4 - Prob. 30ESCh. 8.4 - Find an inverse for 210 modulo 13. Find appositive...Ch. 8.4 - Find an inverse for 41 modulo 660. Find the least...Ch. 8.4 - Prob. 33ESCh. 8.4 - Prob. 34ESCh. 8.4 - Prob. 35ESCh. 8.4 - In 36,37,39 and 40, use the RSA cipher with public...Ch. 8.4 - Prob. 37ESCh. 8.4 - Find the least positive inverse for 43 modulo 660.Ch. 8.4 - Prob. 39ESCh. 8.4 - Prob. 40ESCh. 8.4 - Prob. 41ESCh. 8.4 - Prob. 42ESCh. 8.4 - Prob. 43ESCh. 8.5 - Prob. 1TYCh. 8.5 - Prob. 2TYCh. 8.5 - Prob. 3TYCh. 8.5 - Prob. 4TYCh. 8.5 - Prob. 5TYCh. 8.5 - Prob. 6TYCh. 8.5 - Prob. 7TYCh. 8.5 - Prob. 8TYCh. 8.5 - Prob. 9TYCh. 8.5 - Prob. 10TYCh. 8.5 - Each of the following is a relation on {0,1,2,3}...Ch. 8.5 - Prob. 2ESCh. 8.5 - Let S be the set of all strings of a’s and b’s....Ch. 8.5 - Prob. 4ESCh. 8.5 - Prob. 5ESCh. 8.5 - Let P be the set of all people who have ever lived...Ch. 8.5 - Prob. 7ESCh. 8.5 - Prob. 8ESCh. 8.5 - Prob. 9ESCh. 8.5 - Suppose R and S are antisymmetric relations on a...Ch. 8.5 - Let A={a,b}, and supposeAhas the partial order...Ch. 8.5 - Prob. 12ESCh. 8.5 - Let A={a,b} . Describe all partial order relations...Ch. 8.5 - Let A={a,b,c}. Describe all partial order...Ch. 8.5 - Prob. 15ESCh. 8.5 - Consider the “divides” relation on each of the...Ch. 8.5 - Prob. 17ESCh. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Consider the “divides” relation defined on the set...Ch. 8.5 - Prob. 22ESCh. 8.5 - Prob. 23ESCh. 8.5 - Prob. 24ESCh. 8.5 - Prob. 25ESCh. 8.5 - Prob. 26ESCh. 8.5 - Prob. 27ESCh. 8.5 - Prob. 28ESCh. 8.5 - Prob. 29ESCh. 8.5 - Prob. 30ESCh. 8.5 - Prob. 31ESCh. 8.5 - Prob. 32ESCh. 8.5 - Consider the set A={12,24,48,3,9} ordered by the...Ch. 8.5 - Suppose that R is a partial order relation on a...Ch. 8.5 - Prob. 35ESCh. 8.5 - The set A={2,4,3,6,12,18,24} is partially ordered...Ch. 8.5 - Find a chain of length 2 for the relation defined...Ch. 8.5 - Prob. 38ESCh. 8.5 - Prob. 39ESCh. 8.5 - Prob. 40ESCh. 8.5 - Prob. 41ESCh. 8.5 - Prob. 42ESCh. 8.5 - Prob. 43ESCh. 8.5 - Prob. 44ESCh. 8.5 - Prob. 45ESCh. 8.5 - Prob. 46ESCh. 8.5 - Prob. 47ESCh. 8.5 - Prob. 48ESCh. 8.5 - Prob. 49ESCh. 8.5 - A set S of jobs can be ordered by writing x_y to...Ch. 8.5 - Suppose the tasks described in Example 8.5.12...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- 9arrow_forward8arrow_forward74. Geometry of implicit differentiation Suppose x and y are related 0. Interpret the solution of this equa- by the equation F(x, y) = tion as the set of points (x, y) that lie on the intersection of the F(x, y) with the xy-plane (z = 0). surface Z = a. Make a sketch of a surface and its intersection with the xy-plane. Give a geometric interpretation of the result that dy dx = Fx F χ y b. Explain geometrically what happens at points where F = 0. yarrow_forward
- Example 3.2. Solve the following boundary value problem by ADM (Adomian decomposition) method with the boundary conditions მი მი z- = 2x²+3 дг Əz w(x, 0) = x² - 3x, θω (x, 0) = i(2x+3). ayarrow_forward6. A particle moves according to a law of motion s(t) = t3-12t2 + 36t, where t is measured in seconds and s is in feet. (a) What is the velocity at time t? (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) What is the acceleration at time t? (f) What is the acceleration after 3 s?arrow_forwardpls help asaparrow_forward
- Q1.4 1 Point V=C(R), the vector space of all real-valued continuous functions whose domain is the set R of all real numbers, and H is the subset of C(R) consisting of all of the constant functions. (e.g. the function ƒ : R → R defined by the formula f(x) = 3 for all x E R is an example of one element of H.) OH is a subspace of V. H is not a subspace of V. Save Answerarrow_forwardSolve the following LP problem using the Extreme Point Theorem: Subject to: Maximize Z-6+4y 2+y≤8 2x + y ≤10 2,y20 Solve it using the graphical method. Guidelines for preparation for the teacher's questions: Understand the basics of Linear Programming (LP) 1. Know how to formulate an LP model. 2. Be able to identify decision variables, objective functions, and constraints. Be comfortable with graphical solutions 3. Know how to plot feasible regions and find extreme points. 4. Understand how constraints affect the solution space. Understand the Extreme Point Theorem 5. Know why solutions always occur at extreme points. 6. Be able to explain how optimization changes with different constraints. Think about real-world implications 7. Consider how removing or modifying constraints affects the solution. 8. Be prepared to explain why LP problems are used in business, economics, and operations research.arrow_forwardConstruct a table and find the indicated limit. √√x+2 If h(x) = then find lim h(x). X-8 X-8 Complete the table below. X 7.9 h(x) 7.99 7.999 8.001 8.01 8.1 (Type integers or decimals rounded to four decimal places as needed.)arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Algebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal LittellAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,

Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage

College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning

Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,

Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Algebraic Complexity with Less Relations; Author: The University of Chicago;https://www.youtube.com/watch?v=ZOKM1JPz650;License: Standard Youtube License
Strassen's Matrix Multiplication - Divide and Conquer - Analysis of Algorithm; Author: Ekeeda;https://www.youtube.com/watch?v=UnpySHwAJsQ;License: Standard YouTube License, CC-BY
Trigonometric Equations with Complex Numbers | Complex Analysis #6; Author: TheMathCoach;https://www.youtube.com/watch?v=zdD8Dab1T2Y;License: Standard YouTube License, CC-BY