Cat and Claire are having a conversation about Cat's favorite number.
    
Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits, $\overline{AB}$, such that $A$ and $B$ are both factors of $\overline{AB}$."
    
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"

Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"
    
Claire says, "Now I know your favorite number!"
    
What is Cat's favorite number?

[i]Proposed by Andrew Wu[/i]
1. **Identify the possible numbers:**
   We need to find all two-digit positive integers $\overline{AB}$ such that $A$ and $B$ are both factors of $\overline{AB}$. Let's list them:
   - For $\overline{AB} = 12$, $A = 1$ and $B = 2$ are factors of 12.
   - For $\overline{AB} = 15$, $A = 1$ and $B = 5$ are factors of 15.
   - For $\overline{AB} = 24$, $A = 2$ and $B = 4$ are factors of 24.
   - For $\overline{AB} = 36$, $A = 3$ and $B = 6$ are factors of 36.
   - For $\overline{AB} = 48$, $A = 4$ and $B = 8$ are factors of 48.

   Therefore, the possible numbers are: $12, 15, 24, 36, 48$.

2. **Identify the set of four numbers:**
   Claire mentions that among four numbers, you can start with any one of them, add a second, subtract a third, and get the fourth. Let's test the combinations:
   - $12 + 48 - 36 = 24$
   - $12 + 48 - 24 = 36$
   - $24 + 36 - 12 = 48$
   - $36 + 12 - 24 = 24$

   The set of four numbers that satisfy this condition is $\{12, 24, 36, 48\}$.

3. **Identify Cat's favorite number:**
   Cat's favorite number is among the four numbers and its square is the product of two of the other numbers. Let's test each number:
   - For $12$: $12^2 = 144$, which is not the product of any two numbers in the set.
   - For $24$: $24^2 = 576$, and $576 = 12 \times 48$.
   - For $36$: $36^2 = 1296$, which is not the product of any two numbers in the set.
   - For $48$: $48^2 = 2304$, which is not the product of any two numbers in the set.

   Therefore, Cat's favorite number is $24$ because $24^2 = 12 \times 48$.

The final answer is $\boxed{24}$