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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.

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 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 $b$.