Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.
Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.
Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.
Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.
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.
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.
To begin the Euclidean algorithm for gcd(1701,3768), place the larger number (3768) on the left side of the division equation and the smaller number (1701) as the divisor. Write 3768=1701⋅q+r.
Conditions: The inputs are positive integers.; The larger number is used first on the left-hand side.; The quotient is an integer and the remainder satisfies 0≤r<1701.
To begin the Euclidean algorithm for gcd(1701,3768), place the larger number (3768) on the left side of the division equation and the smaller number (1701) as the divisor. Write 3768=1701⋅q+r.
Conditions: The inputs are positive integers.; The larger number is used first on the left-hand side.; The quotient is an integer and the remainder satisfies 0≤r<1701.