Fundamental Theorem of Arithmetic and Irrational Numbers
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Fundamental Theorem of Arithmetic and Irrational Numbers

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Questions and Answers

What is the HCF of the numbers 6 and 20?

  • 2 (correct)
  • 4
  • 10
  • 6

    • What is the LCM of the numbers 6 and 20?

  • 60 (correct)
  • 40
  • 30
  • 120

    • Which statement about the relationship between HCF and LCM is correct?

  • HCF(a, b) = LCM(a, b) in all cases
  • HCF(a, b) is always greater than LCM(a, b)
  • HCF(a, b) × LCM(a, b) = a × b (correct)
  • HCF(a, b) + LCM(a, b) = a + b

    • Using the prime factorization method, what is the prime factorization of 96?

    <p>2^5 × 3</p> Signup and view all the answers

    How would you find the LCM of 96 and 404 using their HCF?

    <p>Divide the product of 96 and 404 by the HCF</p> Signup and view all the answers

    What is the HCF of the numbers 6, 72, and 120?

    <p>6</p> Signup and view all the answers

    Which of the following is true regarding the product of three numbers and their HCF and LCM?

    <p>The product of three numbers does not equal the product of their HCF and LCM</p> Signup and view all the answers

    The prime factorization of 404 is represented as?

    <p>2^2 × 101</p> Signup and view all the answers

    Who is credited with the first correct proof of the Fundamental Theorem of Arithmetic?

    <p>Carl Friedrich Gauss</p> Signup and view all the answers

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

    <p>They can only be factorised using prime numbers.</p> Signup and view all the answers

    In what manner is the prime factorisation expressed according to the Fundamental Theorem of Arithmetic?

    <p>In ascending order of primes.</p> Signup and view all the answers

    What would the prime factorisation of 32760 look like if expressed in powers of primes?

    <p>2^3 × 3^2 × 5 × 7 × 13</p> Signup and view all the answers

    Which of the following statements is true regarding the uniqueness of prime factorisation?

    <p>The order of primes does not affect the uniqueness of the factorization.</p> Signup and view all the answers

    If a number can be expressed as 4n, what can be concluded about its prime factors?

    <p>It contains only the prime number 2.</p> Signup and view all the answers

    Which mathematician is commonly referred to as the ‘Prince of Mathematicians’?

    <p>Carl Friedrich Gauss</p> Signup and view all the answers

    What major implication does the Fundamental Theorem of Arithmetic have on composite numbers?

    <p>There is exactly one prime factorisation for them, ignoring the order.</p> Signup and view all the answers

    What does the Fundamental Theorem of Arithmetic state regarding prime factors of a number?

    <p>Every integer greater than 1 can be factored into a unique product of prime numbers.</p> Signup and view all the answers

    What assumption leads to a contradiction in the proof of the irrationality of √2?

    <p>That √2 equals a rational number expressed as a fraction.</p> Signup and view all the answers

    In the proof that √2 is irrational, which step is crucial after establishing that 2 divides a²?

    <p>Writing a as a multiple of 2.</p> Signup and view all the answers

    What is the prime factorization of 140?

    <p>2 × 2 × 5 × 7</p> Signup and view all the answers

    What conclusion is made about integers a and b in the proof regarding √3 being irrational?

    <p>They are coprime if they share no common factors other than 1.</p> Signup and view all the answers

    For the integers 26 and 91, what are the LCM and HCF?

    <p>LCM = 182, HCF = 1</p> Signup and view all the answers

    What can be inferred if b² is found to be divisible by 3 in the proof for √3's rationality?

    <p>That 3 is a factor of b.</p> Signup and view all the answers

    When using the prime factorization method, what are the prime factors of 12, 15, and 21?

    <p>2^2 × 3 × 5 × 7</p> Signup and view all the answers

    What method is used to prove that √2 is irrational?

    <p>Proof by contradiction.</p> Signup and view all the answers

    What can be concluded about the product of LCM and HCF from two numbers?

    <p>It equals the product of the two numbers</p> Signup and view all the answers

    Why is it important to express integers a and b as coprime in the context of these proofs?

    <p>To eliminate extraneous factors from consideration.</p> Signup and view all the answers

    What is a key outcome of the proof that both a and b must be divisible by 2?

    <p>It confirms that a and b have common factors larger than 1, contradicting their coprimeness.</p> Signup and view all the answers

    Why is the number 5 irrational?

    <p>It cannot be expressed in the form p/q</p> Signup and view all the answers

    Given HCF(306, 657) = 9, what is LCM(306, 657)?

    <p>2042</p> Signup and view all the answers

    If Sonia takes 18 minutes and Ravi takes 12 minutes to complete a circular path, after how many minutes will they meet again at the starting point?

    <p>36 minutes</p> Signup and view all the answers

    Which of the following claims is true regarding the composite numbers mentioned?

    <p>They have divisors other than 1 and themselves</p> Signup and view all the answers

    What does Euclid's division algorithm state about dividing a positive integer?

    <p>It can be divided by another positive integer leaving a remainder smaller than the divisor.</p> Signup and view all the answers

    What is a unique characteristic of composite numbers according to the Fundamental Theorem of Arithmetic?

    <p>They can be expressed uniquely as products of prime numbers.</p> Signup and view all the answers

    How can the Fundamental Theorem of Arithmetic help determine the nature of a decimal expansion?

    <p>By analyzing the prime factorization of the denominator.</p> Signup and view all the answers

    Which of the following is NOT an application of the Fundamental Theorem of Arithmetic?

    <p>Calculating the highest common factor (HCF) of two numbers.</p> Signup and view all the answers

    What type of numbers were previously studied in Class IX that relate to the Fundamental Theorem of Arithmetic?

    <p>Irrational numbers such as $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$.</p> Signup and view all the answers

    What does the remainder represent in the context of Euclid's division algorithm?

    <p>The part of the dividend not divisible by the divisor.</p> Signup and view all the answers

    Which expression satisfies the condition of Euclid's division algorithm for positive integers?

    <p>If $a = 15$ and $b = 4$, then $r = 3$.</p> Signup and view all the answers

    When is the decimal expansion of a rational number non-terminating repeating?

    <p>When the denominator has prime factors other than 2 or 5.</p> Signup and view all the answers

    Study Notes

    Fundamental Theorem of Arithmetic

    • Every composite (non-prime) number can be uniquely factored into a product of prime numbers, ignoring the order of the factors.
    • Example: 32760 = 2³ × 3² × 5 × 7 × 13
    • Can be used to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more integers.

    Euclid's Division Algorithm

    • Every positive integer a can be divided by any positive integer b to obtain a remainder r, which is smaller than b.
    • It can be used to compute the HCF of two positive integers.

    Irrational Numbers

    • A number s is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0.
    • √2, √3, √5, and √p (where p is a prime) are irrational numbers.
    • Proof by contradiction is used to demonstrate the irrationality of these numbers.

    Applications of the Fundamental Theorem of Arithmetic

    • Proving the irrationality of numbers like √2, √3, and √5.
    • Determining whether the decimal expansion of a rational number is terminating or non-terminating repeating.
    • The prime factorisation of the denominator of a rational number reveals the nature of its decimal expansion.

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    Description

    Explore the Fundamental Theorem of Arithmetic, Euclid's Division Algorithm, and the nature of irrational numbers. This quiz covers unique factorization of composite numbers, computing highest common factors, and the proof of irrationality. Test your understanding and ability to apply these concepts in various mathematical scenarios.

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