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Questions and Answers
Which set represents the natural numbers?
Which set represents the natural numbers?
- $ extbf{ℝ}$
- $ extbf{ℕ}$ (correct)
- $ extbf{ℤ}$
- $ extbf{ℚ}$
If 2 divides 6, which statement is true?
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?
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?
What property states that every nonempty set of natural numbers has a least element?
Which of the following is a correct statement regarding divisibility?
Which of the following is a correct statement regarding divisibility?
Which of the following integers can divide zero?
Which of the following integers can divide zero?
What is the smallest element of the set of positive integers?
What is the smallest element of the set of positive integers?
Which integers are the divisors of 17?
Which integers are the divisors of 17?
What is the greatest common divisor (gcd) of 343 and 280?
What is the greatest common divisor (gcd) of 343 and 280?
Which equation represents the process of finding integers x and y using the Euclidean Algorithm?
Which equation represents the process of finding integers x and y using the Euclidean Algorithm?
What integers x and y satisfy the equation 343x + 280y = 7?
What integers x and y satisfy the equation 343x + 280y = 7?
According to Bezout's Theorem, if (a, b) = d, what can be concluded?
According to Bezout's Theorem, if (a, b) = d, what can be concluded?
In the process of finding x and y using quotients from the Euclidean Algorithm, what is the value of q3 in the given example?
In the process of finding x and y using quotients from the Euclidean Algorithm, what is the value of q3 in the given example?
If d|ab and (d, a) = 1, which statement is correct according to the corollary?
If d|ab and (d, a) = 1, which statement is correct according to the corollary?
During which step of the Euclidean Algorithm does the remainder first become 7?
During which step of the Euclidean Algorithm does the remainder first become 7?
What is the pattern used to derive the values of integers x and y from the quotients in the Euclidean Algorithm?
What is the pattern used to derive the values of integers x and y from the quotients in the Euclidean Algorithm?
What can be concluded if 11 divides both 66 and 198?
What can be concluded if 11 divides both 66 and 198?
If d is a divisor of both a and b, what can be definitely stated about c in terms of a + b?
If d is a divisor of both a and b, what can be definitely stated about c in terms of a + b?
How is the greatest common divisor (gcd) defined?
How is the greatest common divisor (gcd) defined?
Which statement is true regarding the existence of the gcd of two integers?
Which statement is true regarding the existence of the gcd of two integers?
In the example of having 100 coins made up of pennies, dimes, and quarters totaling exactly $5.00, what conclusion can be drawn?
In the example of having 100 coins made up of pennies, dimes, and quarters totaling exactly $5.00, what conclusion can be drawn?
If d divides 4 and 14, what can be the possible values of d?
If d divides 4 and 14, what can be the possible values of d?
What can be said about the gcd of two integers where one of them is prime?
What can be said about the gcd of two integers where one of them is prime?
What is the result of the expression 5(21) - 3(33) in terms of divisibility?
What is the result of the expression 5(21) - 3(33) in terms of divisibility?
What is the result of (n, 1) where n is any positive integer?
What is the result of (n, 1) where n is any positive integer?
What is the greatest common divisor of a positive integer a and 2a?
What is the greatest common divisor of a positive integer a and 2a?
If (a, b) = d, what is the relationship between (a/d, b/d) and 1?
If (a, b) = d, what is the relationship between (a/d, b/d) and 1?
What can be said about the integers q and r in the Division Algorithm?
What can be said about the integers q and r in the Division Algorithm?
What is the value of (n, 0) for any integer n?
What is the value of (n, 0) for any integer n?
Which pair of integers is an example of relatively prime numbers?
Which pair of integers is an example of relatively prime numbers?
What guarantees that the integers q and r from the Division Algorithm are unique?
What guarantees that the integers q and r from the Division Algorithm are unique?
If d and n are any integers, what is the greatest common divisor of (d, nd)?
If d and n are any integers, what is the greatest common divisor of (d, nd)?
In the expression $a = bq + r$, what does 'r' represent?
In the expression $a = bq + r$, what does 'r' represent?
Given $a = -50$ and $b = 8$, what are the values of q and r respectively?
Given $a = -50$ and $b = 8$, what are the values of q and r respectively?
What guarantees the end of the sequence of remainders in the Euclidean Algorithm?
What guarantees the end of the sequence of remainders in the Euclidean Algorithm?
If $gcd(a, b) = d$, which of the following statements is true according to Bezout's Theorem?
If $gcd(a, b) = d$, which of the following statements is true according to Bezout's Theorem?
What can be concluded if both $a$ and $b$ are negative in the context of gcd?
What can be concluded if both $a$ and $b$ are negative in the context of gcd?
What are the values of q and r for the case $a = 75$ and $b = 24$?
What are the values of q and r for the case $a = 75$ and $b = 24$?
What condition must be upheld for the remainder r in the division $a = bq + r$?
What condition must be upheld for the remainder r in the division $a = bq + r$?
What is the first step in applying the Euclidean Algorithm to calculate gcd?
What is the first step in applying the Euclidean Algorithm to calculate gcd?
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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.
- Method 1: Use the remainders in the Euclidean Algorithm:
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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