Euclidean Algorithm and Bezout's Theorem
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Euclidean Algorithm and Bezout's Theorem

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Questions and Answers

Which set represents the natural numbers?

  • $ extbf{ℝ}$
  • $ extbf{ℕ}$ (correct)
  • $ extbf{ℤ}$
  • $ extbf{ℚ}$

    • If 2 divides 6, which statement is true?

  • 6 does not contain 2.
  • 6 is a factor of 2.
  • 2 is a multiple of 6.
  • 2 is a divisor of 6. (correct)

    • Why is the set of all integers ℤ not well ordered?

  • It contains negative integers.
  • It has no smallest element. (correct)
  • It is empty.
  • It is bounded above.

    • What property states that every nonempty set of natural numbers has a least element?

    <p>Well-Ordering Principle</p> Signup and view all the answers

    Which of the following is a correct statement regarding divisibility?

    <p>If $a|b$, there exists integer $c$ such that $b = ac$.</p> Signup and view all the answers

    Which of the following integers can divide zero?

    <p>All integers including zero</p> Signup and view all the answers

    What is the smallest element of the set of positive integers?

    <p>1</p> Signup and view all the answers

    Which integers are the divisors of 17?

    <p>$ ext{±1, ±17}$</p> Signup and view all the answers

    What is the greatest common divisor (gcd) of 343 and 280?

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

    Which equation represents the process of finding integers x and y using the Euclidean Algorithm?

    <p>7 = 343(9) - 280(11)</p> Signup and view all the answers

    What integers x and y satisfy the equation 343x + 280y = 7?

    <p>9 and -11</p> Signup and view all the answers

    According to Bezout's Theorem, if (a, b) = d, what can be concluded?

    <p>There exist integers x and y such that a x + b y = d.</p> Signup and view all the answers

    In the process of finding x and y using quotients from the Euclidean Algorithm, what is the value of q3 in the given example?

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

    If d|ab and (d, a) = 1, which statement is correct according to the corollary?

    <p>d must divide b.</p> Signup and view all the answers

    During which step of the Euclidean Algorithm does the remainder first become 7?

    <p>From 28 to 7</p> Signup and view all the answers

    What is the pattern used to derive the values of integers x and y from the quotients in the Euclidean Algorithm?

    <p>Each subsequent value is a product of the previous value multiplied by a quotient.</p> Signup and view all the answers

    What can be concluded if 11 divides both 66 and 198?

    <p>11 divides 66 and 66 divides 198.</p> Signup and view all the answers

    If d is a divisor of both a and b, what can be definitely stated about c in terms of a + b?

    <p>c is a divisor of a + b.</p> Signup and view all the answers

    How is the greatest common divisor (gcd) defined?

    <p>It is the largest number that divides both a and b without leaving a remainder.</p> Signup and view all the answers

    Which statement is true regarding the existence of the gcd of two integers?

    <p>The gcd is not defined if either integer is zero.</p> Signup and view all the answers

    In the example of having 100 coins made up of pennies, dimes, and quarters totaling exactly $5.00, what conclusion can be drawn?

    <p>It is impossible to have 100 coins worth exactly $5.00.</p> Signup and view all the answers

    If d divides 4 and 14, what can be the possible values of d?

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

    What can be said about the gcd of two integers where one of them is prime?

    <p>It is either 1 or the prime number itself.</p> Signup and view all the answers

    What is the result of the expression 5(21) - 3(33) in terms of divisibility?

    <p>The result is divisible by 3.</p> Signup and view all the answers

    What is the result of (n, 1) where n is any positive integer?

    <p>1</p> Signup and view all the answers

    What is the greatest common divisor of a positive integer a and 2a?

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

    If (a, b) = d, what is the relationship between (a/d, b/d) and 1?

    <p>(a/d, b/d) = 1</p> Signup and view all the answers

    What can be said about the integers q and r in the Division Algorithm?

    <p>They are unique integers.</p> Signup and view all the answers

    What is the value of (n, 0) for any integer n?

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

    Which pair of integers is an example of relatively prime numbers?

    <p>(25, 42)</p> Signup and view all the answers

    What guarantees that the integers q and r from the Division Algorithm are unique?

    <p>The remainder must be less than b.</p> Signup and view all the answers

    If d and n are any integers, what is the greatest common divisor of (d, nd)?

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

    In the expression $a = bq + r$, what does 'r' represent?

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

    Given $a = -50$ and $b = 8$, what are the values of q and r respectively?

    <p>-7 and 6</p> Signup and view all the answers

    What guarantees the end of the sequence of remainders in the Euclidean Algorithm?

    <p>A remainder eventually becomes zero.</p> Signup and view all the answers

    If $gcd(a, b) = d$, which of the following statements is true according to Bezout's Theorem?

    <p>There exist integers x and y such that $ax + by = d$.</p> Signup and view all the answers

    What can be concluded if both $a$ and $b$ are negative in the context of gcd?

    <p>The gcd remains unchanged as $(|a|, |b|)$.</p> Signup and view all the answers

    What are the values of q and r for the case $a = 75$ and $b = 24$?

    <p>4 and 3</p> Signup and view all the answers

    What condition must be upheld for the remainder r in the division $a = bq + r$?

    <p>$0 eq r$ and $r &lt; b$.</p> Signup and view all the answers

    What is the first step in applying the Euclidean Algorithm to calculate gcd?

    <p>Set up the equation $a = bq + r$.</p> Signup and view all the answers

    Study Notes

    Euclidean Algorithm

    • The Euclidean Algorithm is a method for determining the greatest common divisor (GCD) of two integers, denoted as (𝑎, 𝑏) or 𝑔𝑐𝑑(𝑎, 𝑏).
    • The algorithm works by repeatedly finding the remainders of divisions until a remainder of 0 is reached.
    • The final non-zero remainder is the GCD of the original two integers.
    • Example:
      • 343 = 280(1) + 63
      • 280 = 63(4) + 28
      • 63 = 28(2) + 7
      • 28 = 7(4) + 0
    • Therefore, gcd(343, 280) = 7.

    Bezout's Theorem

    • Bezout's Theorem states that for two integers a and b, there exist integers x and y such that ax + by = d, where d is the greatest common divisor of a and b.
    • This theorem can be proven by working the Euclidean algorithm backwards, expressing the GCD (d) as a linear combination of the original integers.
    • To find x and y, we can use two methods:
      • Method 1: Use the remainders in the Euclidean Algorithm:
        • Express each remainder as the difference of two integers in the corresponding equation.
        • Substitute the remainder in the previous equation with this expression and continue this process until you reach the equation where the GCD (d) is expressed as a linear combination of the original integers.
      • Method 2: Use the quotients in the Euclidean Algorithm:
        • Create a table with the quotients from the divisions in the Euclidean Algorithm, starting with q1 and going down to qn, where qn is the final quotient.
        • Calculate the s-values and t-values for each step, starting from the bottom of the table.
        • For the s-values, use the formula: sn = qn-1 * sn-1 + sn-2, where qn-1 is the quotient used in the step.
        • For the t-values, use the formula: tn = qn-1 * tn-1 + tn-2.
        • The final s-value, 𝑠n, will be x, and the final t-value, 𝑡n, will be y.

    Examples

    • Example (Method 1):
      • Find x and y such that 343x + 280y = gcd(343, 280).
      • Using the remainders in the Euclidean Algorithm:
        • 7 = 63 - 28(2)
        • 7 = 63 - (2)(280 - 63(4)) = 63(9) - 280(2)
        • 7 = 9(343 - 280) - (2)280 = 343(9) - 280(11)
      • Therefore, 7 = 343(9) + 280(-11), and we can take x = 9 and y = -11.
    • Example (Method 2):
      • Find x and y such that 343x + 280y = gcd(343, 280).
      • Using the quotients from the Euclidean Algorithm:
        • q1 = 1, q2 = 4, q3 = 2, q4 = 4
      • Create a table with s-values and t-values:
        q1 q2 q3 q4
        0 1 4 9
        0 1 5 11
      • Therefore, x = 9 and y = -11.

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    Description

    This quiz covers the Euclidean Algorithm and Bezout's Theorem. You'll learn how to determine the greatest common divisor (GCD) of two integers and understand the significance of Bezout's identity in number theory. Test your knowledge on these fundamental concepts in mathematics.

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