Real Numbers Chapter 1: Introduction and Properties
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Questions and Answers

What is the Fundamental Theorem of Arithmetic?

  • It states that every composite number can be expressed as a product of primes in a unique way. (correct)
  • It deals with the multiplication of positive integers.
  • It deals with the division of integers.
  • It states that any positive integer can be divided by another positive integer in such a way that it leaves a remainder smaller than the divisor.

    • What is the main purpose of Euclid's division algorithm?

  • To find the quotient and remainder when one positive integer is divided by another. (correct)
  • To determine whether a number is prime or composite.
  • To compute the GCD (Greatest Common Divisor) of two positive integers.
  • To generate a sequence of prime numbers.

    • What does the Fundamental Theorem of Arithmetic state about composite numbers?

  • They can be expressed as a product of primes in multiple ways.
  • They are only divisible by 1 and themselves.
  • They can be expressed as a product of primes in a unique way. (correct)
  • They can only be divided by even numbers.

    • What is the connection between Euclid's division algorithm and computing the HCF (Highest Common Factor) of two positive integers?

    <p>The number obtained as the remainder in Euclid's division algorithm is used to compute the HCF.</p> Signup and view all the answers

    Which property of positive integers does Euclid's division algorithm primarily deal with?

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

    What significant application does the Fundamental Theorem of Arithmetic have in mathematics?

    <p>It is used in cryptography and prime factorization.</p> Signup and view all the answers

    What is the primary purpose of using the Fundamental Theorem of Arithmetic?

    <p>To explore the nature of the decimal expansion of rational numbers</p> Signup and view all the answers

    Can any natural number be obtained by multiplying prime numbers?

    <p>No, not all natural numbers can be obtained by multiplying prime numbers</p> Signup and view all the answers

    What can be inferred about the size of the collection obtained by combining all possible primes?

    <p>The collection contains infinitely many integers</p> Signup and view all the answers

    What does factorization of a number involve?

    <p>Expressing it as the product of its prime factors</p> Signup and view all the answers

    What is the purpose of using a factor tree for factorization?

    <p>To visually represent the prime factorization of a number</p> Signup and view all the answers

    What does the Fundamental Theorem of Arithmetic state about composite numbers?

    <p>Every composite number has a unique prime factorization</p> Signup and view all the answers

    What type of application does the Fundamental Theorem of Arithmetic have in mathematics?

    <p>It is used to prove properties related to the factorization of positive integers</p> Signup and view all the answers

    What can be concluded about the relationship between Euclid's division algorithm and the Fundamental Theorem of Arithmetic?

    <p>The Fundamental Theorem of Arithmetic is a generalization of Euclid's division algorithm</p> Signup and view all the answers

    What is significant about using the Fundamental Theorem of Arithmetic to explore decimal expansions?

    <p>'Fundamental Theorem of Arithmetic' allows us to determine when rational numbers have terminating or repeating decimal expansions</p> Signup and view all the answers

    'Can any natural number be obtained by multiplying prime numbers?' In light of this question, what conclusion can be drawn about the behavior of natural numbers when multiplied by primes?

    <p>'Natural numbers when multiplied by primes may or may not result in another natural number'</p> Signup and view all the answers

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