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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 $(n, 0)$.

Replacing many subtractions by division simplifies the algorithm by using the remainder directly. For $b=aq+r$ with $0 \le r < a$, the pair $(a,b)$ is replaced by $(r,a)$.

Conditions: $a$ and $b$ are nonnegative integers.; $b = aq + r$ where $0 \le r < a$.; The algorithm proceeds by replacing the pair $(a,b)$ with $(r,a)$.

Replacing many subtractions by division simplifies the algorithm by using the remainder directly. For $b=aq+r$ with $0 \le r < a$, the pair $(a,b)$ is replaced by $(r,a)$.

Conditions: $a$ and $b$ are nonnegative integers.; $b = aq + r$ where $0 \le r < a$.; The algorithm proceeds by replacing the pair $(a,b)$ with $(r,a)$.