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The Euclidean algorithm computes a greatest common divisor by repeated division with remainder. Replacing (a,b) by (b,r) preserves the common divisors, and the strictly smaller nonnegative remainders force termination.
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From 35 to 202 seconds, the source develops divisibility lemmas and repeatedly replaces a pair by a remainder pair to explain GCD preservation. The source shows only one direction of the common-divisor argument; the public editorial note supplies the converse, so the link is classified as explanation rather than proof.
From 14 to 84 seconds, the source computes and , then explains that a common divisor divides the difference. The reviewed material explicitly states that the converse preservation step is absent, so this is not classified as a complete proof.
From 25 to 245 seconds, the source carries two positive-integer remainder chains to a zero remainder: , gives gcd(10,45)=5; the seven-step 3768 and 1701 calculation ends at gcd=3. The source demonstrates the algorithm but does not prove its invariant or termination, so the role is application rather than proof.
From 0 to 163 seconds, the board recalls repeated division, works all eight divisions for 5295 and 4321, reaches a zero remainder, and identifies the preceding nonzero remainder as 1. The video applies the algorithm; it does not prove the general theorem.
From 202 to 282 seconds, the source derives the recursive remainder rule, implements it in Java, and argues that the remainder becomes less than half within at most two calls. The public review limits the complexity claim to remainder-call counts under a unit-cost model and corrects the slide typo near 270 seconds.
From 0 to 123 seconds, the lesson states the repeated-division procedure and stopping rule, gives a short 104 and 84 warm-up, then completes all five divisions for 1785 and 546. It applies the algorithm rather than proving the general theorem.
A reviewed path from functions and finite counting to arrangements, selections, divisibility and the Euclidean algorithm. Every new concept is paired with verified video evidence, while the prerequisite arrows describe this map’s editorial learning order.
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, 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 .
The example ends with 4 because applying the subtraction rule repeatedly yields 4 as the final value. First, , creating the pair 8 and 4.
Conditions: Start with the pair 12 and 8.; Repeatedly subtract the smaller from the larger.
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.
To compute , apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with .
Conditions: The inputs are 5295 and 4321.; The Euclidean algorithm is used.; The division algorithm is applied at each step.
The Euclidean algorithm computes the greatest common divisor by repeatedly applying the division algorithm. Starting with two natural numbers and , you divide the larger by the smaller to get a quotient and a remainder.
Conditions: The inputs and are natural numbers.; The division algorithm is applied repeatedly.; The process stops when a remainder equals 0.
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.
In the displayed Euclidean algorithm, and are the two initial natural numbers whose gcd is being found. represents the quotient at the -th division step.
Conditions: The symbols are from the general statement of the Euclidean algorithm on the left board.