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The base case for the Euclidean algorithm is when the second argument of the pair is zero. At this point, the algorithm stops, and the first argument is returned as the greatest common divisor.

Conditions: The algorithm is in the process of replacing pairs using $\gcd(a,b)=\gcd(b,a\bmod b)$.; The second argument $b$ has reached zero.

The base case for the Euclidean algorithm is when the second argument of the pair is zero. At this point, the algorithm stops, and the first argument is returned as the greatest common divisor.

Conditions: The algorithm is in the process of replacing pairs using $\gcd(a,b)=\gcd(b,a\bmod b)$.; The second argument $b$ has reached zero.

To trace the Euclidean algorithm for $\gcd(10,15)$, repeatedly apply the step $\gcd(a,b)=\gcd(b,a\bmod b)$ until the second argument is zero. The sequence is $\gcd(10,15)=\gcd(15,10)=\gcd(10,5)=\gcd(5,0)=5$.

Conditions: The initial pair is (10, 15).; The algorithm uses the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$.; The process stops when the second argument is zero.

To trace the Euclidean algorithm for $\gcd(10,15)$, repeatedly apply the step $\gcd(a,b)=\gcd(b,a\bmod b)$ until the second argument is zero. The sequence is $\gcd(10,15)=\gcd(15,10)=\gcd(10,5)=\gcd(5,0)=5$.

Conditions: The initial pair is (10, 15).; The algorithm uses the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$.; The process stops when the second argument is zero.

For a nonnegative integer $n$, the greatest common divisor of $n$ and $0$ is $n$ itself. This serves as the base case for the Euclidean algorithm, terminating the recursion when the second argument becomes zero.

Conditions: $n$ is a nonnegative integer.; The pair is $(n, 0)$.

For a nonnegative integer $n$, the greatest common divisor of $n$ and $0$ is $n$ itself. This serves as the base case for the Euclidean algorithm, terminating the recursion when the second argument becomes zero.

Conditions: $n$ is a nonnegative integer.; The pair is $(n, 0)$.

The Euclidean algorithm using division terminates when one of the entries in the current pair $(a,b)$ becomes zero. Specifically, when the remainder $r$ in the step $b=aq+r$ becomes zero, the next pair is $(0,a)$ or $(a,0)$, and the algorithm stops, returning the non-zero entry as the gcd.

Conditions: The algorithm uses the step $\gcd(a,b) = \gcd(b, a \bmod b)$.; $a$ and $b$ are nonnegative integers.; Termination occurs when the second argument is 0.

The Euclidean algorithm using division terminates when one of the entries in the current pair $(a,b)$ becomes zero. Specifically, when the remainder $r$ in the step $b=aq+r$ becomes zero, the next pair is $(0,a)$ or $(a,0)$, and the algorithm stops, returning the non-zero entry as the gcd.

Conditions: The algorithm uses the step $\gcd(a,b) = \gcd(b, a \bmod b)$.; $a$ and $b$ are nonnegative integers.; Termination occurs when the second argument is 0.

Repeating the subtraction operation reduces the magnitude of the numbers involved while preserving their common divisors. Since the numbers are positive integers, they cannot decrease indefinitely.

Conditions: The initial numbers are positive integers.; The operation is subtracting the smaller from the larger.; The goal is to preserve the gcd while reducing the numbers.

Repeating the subtraction operation reduces the magnitude of the numbers involved while preserving their common divisors. Since the numbers are positive integers, they cannot decrease indefinitely.

Conditions: The initial numbers are positive integers.; The operation is subtracting the smaller from the larger.; The goal is to preserve the gcd while reducing the numbers.

Subtracting the smaller positive integer from the larger preserves their common divisors because an integer divides both $a$ and $b$ exactly when it divides both $a$ and $b-a$. Consequently, the sets of common divisors, and hence the greatest common divisor (gcd), stay the same.

Conditions: $a$ and $b$ are positive integers.; $b > a$ (so we subtract $a$ from $b$).

Subtracting the smaller positive integer from the larger preserves their common divisors because an integer divides both $a$ and $b$ exactly when it divides both $a$ and $b-a$. Consequently, the sets of common divisors, and hence the greatest common divisor (gcd), stay the same.

Conditions: $a$ and $b$ are positive integers.; $b > a$ (so we subtract $a$ from $b$).

The Euclidean algorithm reduces the problem size by replacing the larger number with the remainder of the division of the two numbers. Since the remainder $r$ satisfies $0 \le r < a$ (where $a$ is the divisor), the new number is strictly smaller than the previous larger number, ensuring progress towards the base case.

Conditions: The current pair is $(a,b)$ with $b \ge a > 0$.; The step computes $b = aq + r$ with $0 \le r < a$.; The new pair is $(r,a)$ or $(a,r)$ depending on implementation, but effectively the max value decreases.

The Euclidean algorithm reduces the problem size by replacing the larger number with the remainder of the division of the two numbers. Since the remainder $r$ satisfies $0 \le r < a$ (where $a$ is the divisor), the new number is strictly smaller than the previous larger number, ensuring progress towards the base case.

Conditions: The current pair is $(a,b)$ with $b \ge a > 0$.; The step computes $b = aq + r$ with $0 \le r < a$.; The new pair is $(r,a)$ or $(a,r)$ depending on implementation, but effectively the max value decreases.

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).

The greatest common divisor of 10 and 15 is 5. This is determined by listing the common positive divisors of both numbers, which are 1 and 5, and selecting the largest one.

Conditions: The inputs are the positive integers 10 and 15.; The method used is listing all common positive divisors.

The greatest common divisor of 10 and 15 is 5. This is determined by listing the common positive divisors of both numbers, which are 1 and 5, and selecting the largest one.

Conditions: The inputs are the positive integers 10 and 15.; The method used is listing all common positive divisors.

The Euclidean algorithm keeps the GCD by repeatedly applying the step $\gcd(a,b)=\gcd(b,a\bmod b)$ for nonnegative $a$ and positive $b$. This replaces the pair with the divisor and the remainder, preserving the greatest common divisor.

Conditions: $a$ is a nonnegative integer.; $b$ is a positive integer.; The operation $a\bmod b$ computes the remainder of $a$ divided by $b$.

The Euclidean algorithm keeps the GCD by repeatedly applying the step $\gcd(a,b)=\gcd(b,a\bmod b)$ for nonnegative $a$ and positive $b$. This replaces the pair with the divisor and the remainder, preserving the greatest common divisor.

Conditions: $a$ is a nonnegative integer.; $b$ is a positive integer.; The operation $a\bmod b$ computes the remainder of $a$ divided by $b$.

The Euclidean algorithm terminates because the remainder in each step is strictly smaller than the previous divisor. Since the second argument is a nonnegative integer that decreases with each step, it must eventually reach zero.

Conditions: The algorithm operates on nonnegative integers.; Each step replaces the pair $(a,b)$ with $(b, a\bmod b)$.; The remainder $a\bmod b$ is strictly less than $b$ when $b > 0$.

The Euclidean algorithm terminates because the remainder in each step is strictly smaller than the previous divisor. Since the second argument is a nonnegative integer that decreases with each step, it must eventually reach zero.

Conditions: The algorithm operates on nonnegative integers.; Each step replaces the pair $(a,b)$ with $(b, a\bmod b)$.; The remainder $a\bmod b$ is strictly less than $b$ when $b > 0$.

You should stop the Euclidean algorithm when the second argument of the pair becomes zero. At this point, the first argument is the greatest common divisor.

Conditions: The algorithm is tracing the steps of the Euclidean algorithm.; The pair is represented as $(a,b)$.; The operation is $\gcd(a,b)=\gcd(b,a\bmod b)$.

You should stop the Euclidean algorithm when the second argument of the pair becomes zero. At this point, the first argument is the greatest common divisor.

Conditions: The algorithm is tracing the steps of the Euclidean algorithm.; The pair is represented as $(a,b)$.; The operation is $\gcd(a,b)=\gcd(b,a\bmod b)$.