To find the greatest common divisor of two large numbers, repeatedly apply the division-with-remainder step. Start by dividing the larger number by the smaller number.
Conditions: The inputs are two positive integers.; The division algorithm is applied at each step.; The process stops when a remainder equals 0.
To find the greatest common divisor of two large numbers, repeatedly apply the division-with-remainder step. Start by dividing the larger number by the smaller number.
Conditions: The inputs are two positive integers.; The division algorithm is applied at each step.; The process stops when a remainder equals 0.
The Euclidean algorithm computes the greatest common divisor by repeatedly applying the division algorithm. Starting with two natural numbers a and b, you divide the larger by the smaller to get a quotient and a remainder.
Conditions: The inputs a and b are natural numbers.; The division algorithm is applied repeatedly.; The process stops when a remainder equals 0.
The Euclidean algorithm computes the greatest common divisor by repeatedly applying the division algorithm. Starting with two natural numbers a and b, you divide the larger by the smaller to get a quotient and a remainder.
Conditions: The inputs a and b are natural numbers.; The division algorithm is applied repeatedly.; The process stops when a remainder equals 0.
The Euclidean algorithm finds the greatest common divisor (GCD) of two integers without needing to factor them. The process involves repeatedly performing long division: divide the larger number by the smaller number, then divide the previous divisor by the remainder, and continue this process.
Conditions: Applies to two integers.; Requires repeated long division.; Stops when the remainder is zero.
The Euclidean algorithm finds the greatest common divisor (GCD) of two integers without needing to factor them. The process involves repeatedly performing long division: divide the larger number by the smaller number, then divide the previous divisor by the remainder, and continue this process.
Conditions: Applies to two integers.; Requires repeated long division.; Stops when the remainder is zero.
When the Euclidean algorithm yields a greatest common divisor of 1, it means the two input numbers are relatively prime (or coprime). This indicates that they share no common positive integer divisors other than 1.
Conditions: The inputs are natural numbers.; The Euclidean algorithm terminates with a last nonzero remainder of 1.
When the Euclidean algorithm yields a greatest common divisor of 1, it means the two input numbers are relatively prime (or coprime). This indicates that they share no common positive integer divisors other than 1.
Conditions: The inputs are natural numbers.; The Euclidean algorithm terminates with a last nonzero remainder of 1.
The Euclidean algorithm moves the old divisor to the left-hand side (new dividend) and the old remainder to the smaller-number position (new divisor) to recursively reduce the problem. This shift ensures that each subsequent division step operates on smaller numbers while preserving the greatest common divisor of the original pair, continuing until a remainder of zero is reached.
Conditions: The algorithm is applied to two positive integers.; The previous remainder is not zero.; The process continues until a remainder of 0 is obtained.
The Euclidean algorithm moves the old divisor to the left-hand side (new dividend) and the old remainder to the smaller-number position (new divisor) to recursively reduce the problem. This shift ensures that each subsequent division step operates on smaller numbers while preserving the greatest common divisor of the original pair, continuing until a remainder of zero is reached.
Conditions: The algorithm is applied to two positive integers.; The previous remainder is not zero.; The process continues until a remainder of 0 is obtained.
Yes, in this context, "greatest common factor" is being used for what is more commonly called the greatest common divisor in many modern texts. The mathematical procedure shown is the same subtraction-based Euclidean algorithm.
Conditions: Used informally in the explanation of why the algorithm works.
Yes, in this context, "greatest common factor" is being used for what is more commonly called the greatest common divisor in many modern texts. The mathematical procedure shown is the same subtraction-based Euclidean algorithm.
Conditions: Used informally in the explanation of why the algorithm works.
To form the next division line, you take the divisor from the previous line and make it the dividend of the new line. Then, you take the remainder from the previous line and make it the divisor of the new line.
Conditions: You have just completed a division step in the Euclidean algorithm.; The previous remainder is not 0.
To form the next division line, you take the divisor from the previous line and make it the dividend of the new line. Then, you take the remainder from the previous line and make it the divisor of the new line.
Conditions: You have just completed a division step in the Euclidean algorithm.; The previous remainder is not 0.
To start the Euclidean algorithm for gcd(10,45), you write the larger number as the smaller number multiplied by an unknown quotient plus an unknown remainder. Specifically, you set up the division equation 45=10⋅q+r.
Conditions: The inputs are positive integers.; The larger number is placed on the left-hand side of the equation.; The quotient is an integer and the remainder satisfies 0≤r<10.
To start the Euclidean algorithm for gcd(10,45), you write the larger number as the smaller number multiplied by an unknown quotient plus an unknown remainder. Specifically, you set up the division equation 45=10⋅q+r.
Conditions: The inputs are positive integers.; The larger number is placed on the left-hand side of the equation.; The quotient is an integer and the remainder satisfies 0≤r<10.
The speaker verbally says "greatest common denominator," but the mathematical notation on the board is "gcd," which conventionally stands for "greatest common divisor." The context of dividing integers to find a common factor confirms that the intended concept is the greatest common divisor, and the spoken word is a verbal slip.
Conditions: The video discusses finding the common factor of two integers.; The board displays the notation gcd(a;b).; The procedure involves repeated integer division.
The speaker verbally says "greatest common denominator," but the mathematical notation on the board is "gcd," which conventionally stands for "greatest common divisor." The context of dividing integers to find a common factor confirms that the intended concept is the greatest common divisor, and the spoken word is a verbal slip.
Conditions: The video discusses finding the common factor of two integers.; The board displays the notation gcd(a;b).; The procedure involves repeated integer division.
In the Euclidean algorithm, when the division process yields a remainder of zero, the greatest common divisor of the original two integers is the last non-zero remainder obtained. The algorithm stops at this point because the method is over, and the last non-zero remainder is guaranteed to divide both original numbers evenly.
Conditions: The Euclidean algorithm is applied to two integers.; The process of repeated long division is followed until a remainder of zero is reached.
In the Euclidean algorithm, when the division process yields a remainder of zero, the greatest common divisor of the original two integers is the last non-zero remainder obtained. The algorithm stops at this point because the method is over, and the last non-zero remainder is guaranteed to divide both original numbers evenly.
Conditions: The Euclidean algorithm is applied to two integers.; The process of repeated long division is followed until a remainder of zero is reached.
Strictly speaking, the standard stopping condition for the Euclidean algorithm is to continue until the remainder is 0. The last nonzero remainder is then the gcd.
Conditions: The inputs are natural numbers.; The Euclidean algorithm is being applied.
Strictly speaking, the standard stopping condition for the Euclidean algorithm is to continue until the remainder is 0. The last nonzero remainder is then the gcd.
Conditions: The inputs are natural numbers.; The Euclidean algorithm is being applied.