How do I use the Euclidean algorithm to find the greatest common divisor of two large numbers?
Conditions
- The inputs are two positive integers.
- The division algorithm is applied at each step.
- The process stops when a remainder equals 0.
Reasoning, step by step
- Identify the two large numbers, for example, 1701 and 3768.
- Divide the larger number by the smaller number: .
- Move the previous divisor (1701) to the left side and the remainder (366) to the divisor position: .
- Repeat the shift and divide process: , , , .
- Perform the final division: .
- Identify the last nonzero remainder, which is 3, as the greatest common divisor.
Example
The board shows the step-by-step division equations for gcd(1701;3768), ending with a remainder of 0. The speaker draws an arrow from the final remainder '0' to the previous remainder '3', and boxes the '3'.
Common misconceptions
- Trying to factor the large numbers into primes first.
- Stopping the algorithm before the remainder reaches zero.
- Confusing the quotient with the remainder in the shift step.
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Related questions
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 .
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, 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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