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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.