Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)

[i]Proposed by Andrew Wu[/i]
1. Let's denote a two-digit number as $\underline{AB}$, where $A$ is the tens digit and $B$ is the units digit. Therefore, the number can be expressed as $10A + B$.
2. The digital reduction of $\underline{AB}$ is defined as $\underline{AB} - A - B$. Substituting the expression for $\underline{AB}$, we get:
   \[
   \underline{AB} - A - B = (10A + B) - A - B
   \]
3. Simplify the expression:
   \[
   (10A + B) - A - B = 10A - A + B - B = 9A
   \]
4. Therefore, the digital reduction of any two-digit number $\underline{AB}$ is $9A$.
5. Since $A$ is a digit from 1 to 9 (as $A$ is the tens digit of a two-digit number), the possible values of $9A$ are $9, 18, 27, 36, 45, 54, 63, 72, 81$.
6. To find the greatest common divisor (GCD) of these values, observe that each value is a multiple of 9. Therefore, the GCD of these values is 9.

Conclusion:
The greatest common divisor of the digital reductions of all two-digit positive integers is $\boxed{9}$.