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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 $b$).