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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 $b > 0$.

By the definition of division with remainder, for any integers $a$ and $b$ ($b>0$), there exist unique integers $q$ and $r$ such that $a = bq + r$ and $0 \le r < b$. This constraint on $r$ ensures that the remainder is always strictly smaller than the divisor $b$, which is crucial for the algorithm's termination.

Conditions: $b$ is a positive integer.; $a$ is a nonnegative integer.; The operation is $a \bmod b$.

By the definition of division with remainder, for any integers $a$ and $b$ ($b>0$), there exist unique integers $q$ and $r$ such that $a = bq + r$ and $0 \le r < b$. This constraint on $r$ ensures that the remainder is always strictly smaller than the divisor $b$, which is crucial for the algorithm's termination.

Conditions: $b$ is a positive integer.; $a$ is a nonnegative integer.; The operation is $a \bmod b$.

In the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$, the remainder $a\bmod b$ becomes the new second argument of the pair. It replaces the original dividend $a$ in the next iteration.

Conditions: $a$ is a nonnegative integer.; $b$ is a positive integer.; The operation is the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$.

In the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$, the remainder $a\bmod b$ becomes the new second argument of the pair. It replaces the original dividend $a$ in the next iteration.

Conditions: $a$ is a nonnegative integer.; $b$ is a positive integer.; The operation is the Euclidean step $\gcd(a,b)=\gcd(b,a\bmod b)$.