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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.

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).