Modular Arithmetic and Multiplicative Inverses Quiz chapter 2
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Questions and Answers

Which of the following is an example of a publication in which confidentiality of the stored data is the most important requirement?

  • An information system in a pathological laboratory maintaining patient's data (correct)
  • An examination section of a university managing sensitive exam papers
  • A student information system used for maintaining student data in a university
  • A student maintaining a blog to post public information

    • Which of the following is an example of a publication in which data integrity is the most important requirement?

  • An information system in a pathological laboratory maintaining patient's data
  • A student information system used for maintaining student data in a university
  • An examination section of a university managing sensitive exam papers (correct)
  • A student maintaining a blog to post public information

    • Which of the following is an example in which system availability is the most important requirement?

  • An information system in a pathological laboratory maintaining patient's data
  • A student maintaining a blog to post public information
  • An examination section of a university managing sensitive exam papers
  • A university library managing the distribution of books (correct)

    • Which set represents the set of nonnegative integers less than 8?

    <p>Z8</p> Signup and view all the answers

    What is the multiplicative inverse of 3 modulo 8?

    <p>7</p> Signup and view all the answers

    Which property of modular arithmetic is represented by the equation (w * (x + y)) mod n = [(w * x) + (w * y)] mod n?

    <p>Distributive Law</p> Signup and view all the answers

    What is the smallest nonnegative integer to which 23 is congruent modulo 8?

    <p>7</p> Signup and view all the answers

    According to the Euclidean algorithm, what is the greatest common divisor of 710 and 310?

    <p>10</p> Signup and view all the answers

    What is the gcd(a, b) if a = 1160718174 and b = 316258250?

    <p>1078</p> Signup and view all the answers

    What is the remainder when 1160718174 is divided by 316258250?

    <p>211943424</p> Signup and view all the answers

    What is the gcd(r1, r2) if r1 = 104314826 and r2 = 3313772?

    <p>1078</p> Signup and view all the answers

    Which equation represents the definition of modular arithmetic?

    <p>$a \mod n = r$</p> Signup and view all the answers

    What is the congruence relation between 73 and 4 modulo 23?

    <p>$73 \equiv 4 \pmod{23}$</p> Signup and view all the answers

    What is the result of $(11 \mod 8) + (15 \mod 8) \mod 8$?

    <p>2</p> Signup and view all the answers

    What is the multiplicative inverse of 5 modulo 8?

    <p>7</p> Signup and view all the answers

    Which integers have a multiplicative inverse in Z8?

    <p>1, 3, 5, and 7</p> Signup and view all the answers

    What is the greatest common divisor (gcd) of 55 and 22?

    <p>11</p> Signup and view all the answers

    What is the relationship between the set of common divisors of a and b, and the set of common divisors of b and (a mod b)?

    <p>They are equal</p> Signup and view all the answers

    According to the text, what is the division algorithm?

    <p>$a = qn + r$, where $0 \leq r \leq n$ and $q = \lfloor \frac{a}{n} \rfloor$</p> Signup and view all the answers

    According to the text, what is the greatest common divisor (gcd) of two integers?

    <p>The largest integer that divides both integers</p> Signup and view all the answers

    According to the text, when are two integers considered relatively prime?

    <p>When their only common positive integer factor is 1</p> Signup and view all the answers

    According to the text, what is the Euclidean algorithm used for?

    <p>Determining the greatest common divisor of two positive integers</p> Signup and view all the answers

    Which of the following is the definition of divisibility?

    <p>A nonzero integer b divides an integer a if a = mb for some m.</p> Signup and view all the answers

    Which of the following is true about the positive divisors of 24?

    <p>The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.</p> Signup and view all the answers

    Which of the following is true about modular arithmetic?

    <p>Modular arithmetic is an operation on integers.</p> Signup and view all the answers

    What is the gcd(710, 310) according to the Euclidean algorithm?

    <p>The gcd(710, 310) is 10.</p> Signup and view all the answers

    Study Notes

    Publication Requirements

    • Confidentiality of stored data is most important in publications related to personal or sensitive information (e.g., medical records, financial data).
    • Data integrity is most important in publications related to critical systems or applications where data accuracy is crucial (e.g., financial transactions, scientific research).
    • System availability is most important in publications related to real-time systems or applications where downtime is not acceptable (e.g., emergency services, online banking).

    Modular Arithmetic

    • The set of nonnegative integers less than 8 is {0, 1, 2, 3, 4, 5, 6, 7}.
    • The multiplicative inverse of 3 modulo 8 is 3.
    • The equation (w * (x + y)) mod n = [(w * x) + (w * y)] mod n represents the distributive property of modular arithmetic.
    • The smallest nonnegative integer to which 23 is congruent modulo 8 is 7.

    Greatest Common Divisors (GCDs)

    • The GCD of 710 and 310 is 10.
    • The GCD of 1160718174 and 316258250 is 2.
    • The remainder when 1160718174 is divided by 316258250 is 2.
    • The GCD of 104314826 and 3313772 is 2.

    Modular Arithmetic Properties

    • The equation a ≡ b (mod n) is the definition of modular arithmetic.
    • The congruence relation between 73 and 4 modulo 23 is 73 ≡ 4 (mod 23).
    • The result of $(11 \mod 8) + (15 \mod 8) \mod 8$ is 4.
    • The multiplicative inverse of 5 modulo 8 is 5.

    Integers with Multiplicative Inverses

    • The integers with multiplicative inverses in Z8 are 1, 3, 5, and 7.

    Greatest Common Divisors and Divisibility

    • The GCD of 55 and 22 is 11.
    • The set of common divisors of a and b is a subset of the set of common divisors of b and (a mod b).
    • The definition of divisibility is "a divides b" if and only if there exists an integer k such that b = ka.
    • The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

    Euclidean Algorithm

    • The Euclidean algorithm is used to find the GCD of two integers.
    • The GCD of two integers a and b is the largest integer that divides both a and b without leaving a remainder.
    • Two integers are considered relatively prime if their GCD is 1.
    • The division algorithm is a procedure for finding the quotient and remainder of two integers.
    • The Euclidean algorithm is used to find the GCD of two integers by repeatedly applying the division algorithm.

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    Chapter 2: Introduction to Number Theory PDF

    Description

    Test your knowledge on the properties of modular arithmetic and finding multiplicative inverses in this quiz. Learn how to use the multiplication table to find the multiplicative inverse of an integer and understand the concept of modular arithmetic. Explore the set Zn and discover which integers have a multiplicative inverse in mod 8.

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