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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.