3. Alice and Bob are independently trying to figure out a secret password to Cathy's bitcoin wallet. Both of them have already figured out that:
- it is a 4-digit number whose first digit is 5;
- it is a multiple of 9 ;
- The larger number is more likely to be a password than a smaller number.
Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed:
Alice: "I have no idea what the number is."
Bob: "I have no idea too."

After that both of them knew which number they should try first. Identify this number.
Solution. Let the password be $5 X Y Z$. As this number is a multiple of 9 , we know that $5+X+Y+Z$ is a multiple of 9 . Alice knows $5+X+Y$ and since she cannot identify the number it must be that $Z=0$ or $Z=9$. Similarly, $X=0$ or $X=9$. So the largest possibility is 5949 .