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Repeating the subtraction operation reduces the magnitude of the numbers involved while preserving their common divisors. Since the numbers are positive integers, they cannot decrease indefinitely.

Conditions: The initial numbers are positive integers.; The operation is subtracting the smaller from the larger.; The goal is to preserve the gcd while reducing the numbers.

Repeating the subtraction operation reduces the magnitude of the numbers involved while preserving their common divisors. Since the numbers are positive integers, they cannot decrease indefinitely.

Conditions: The initial numbers are positive integers.; The operation is subtracting the smaller from the larger.; The goal is to preserve the gcd while reducing the numbers.

Subtracting the smaller positive integer from the larger preserves their common divisors because an integer divides both $a$ and $b$ exactly when it divides both $a$ and $b-a$. Consequently, the sets of common divisors, and hence the greatest common divisor (gcd), stay the same.

Conditions: $a$ and $b$ are positive integers.; $b > a$ (so we subtract $a$ from $b$).

Subtracting the smaller positive integer from the larger preserves their common divisors because an integer divides both $a$ and $b$ exactly when it divides both $a$ and $b-a$. Consequently, the sets of common divisors, and hence the greatest common divisor (gcd), stay the same.

Conditions: $a$ and $b$ are positive integers.; $b > a$ (so we subtract $a$ from $b$).

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 $b > 0$.