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]