Positive divisor of both $a$ and $a$ + 1 is ________.

Step 1 (Coprimality of Consecutive Integers): Two consecutive integers $a$ and $a+1$ are always coprime. To prove this, let $d$ be a positive integer divisor of both $a$ and $a+1$:

Step 2 (Linear combination property): If $d$ divides two numbers, it also divides their difference:

Step 3 (Solve for d): The only positive integer divisor of 1 is 1 itself. Therefore, the only positive divisor of both $a$ and $a+1$ is 1.

Step 4 (Conclusion): The positive divisor of both $a$ and $a+1$ is exactly 1.