Why does the Euclidean algorithm stop when the remainder is zero, and what is the final answer?
Conditions
- The Euclidean algorithm is applied to two integers.
- The process of repeated long division is followed until a remainder of zero is reached.
Reasoning, step by step
- Perform repeated long division on the two integers.
- Observe the sequence of remainders.
- Stop the process when a division yields a remainder of zero.
- Identify the remainder from the immediately preceding step (the last non-zero remainder).
- Conclude that this last non-zero remainder is the greatest common divisor.
Example
The narrator states, 'When you get a remainder of zero, you stop and the method is over. The last nonzero remainder is the greatest common divisor.' An arrow points to the remainder '21' in the second-to-last division step at 01:51, identifying it as the result.
Common misconceptions
- Believing that the zero remainder itself is the greatest common divisor.
- Thinking the algorithm must continue indefinitely even after reaching a zero remainder.
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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.
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 , 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 .
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 .
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.
In the Euclidean algorithm, the previous divisor becomes the new dividend (placed on the left side of the equation), and the previous remainder becomes the new divisor (placed on the right side). This recursive shift carries the numbers forward so that each step divides the former divisor by the former remainder, continuing until a remainder of zero is reached.
Conditions: The algorithm is applied to positive integers.; The previous remainder is not zero.; The process follows the standard division-with-remainder format .
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