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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 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 ($a \bmod b$).