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.
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.
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.
A common divisor also divides the difference because division is interpreted as repeated subtraction. If a number divides evenly into the larger number and the smaller number, subtracting the smaller number repeatedly from the larger one will eventually leave a difference that the same divisor also divides evenly.
Conditions: There are two numbers in the example.; A chosen divisor divides evenly in the sense described by the speaker.; The subtraction is performed from the larger number using the smaller number.
A common divisor also divides the difference because division is interpreted as repeated subtraction. If a number divides evenly into the larger number and the smaller number, subtracting the smaller number repeatedly from the larger one will eventually leave a difference that the same divisor also divides evenly.
Conditions: There are two numbers in the example.; A chosen divisor divides evenly in the sense described by the speaker.; The subtraction is performed from the larger number using the smaller number.