A math teacher thought of a positive integer of two digits. A math teacher thought of a positive integer of two
digits. She wants her two intelligent students Hanna
and Charlie determine the exact number thought.
For this, Hanna is privately told how many positive divisors the number has; Charlie, the sum of the digits of the number.
A brief conversation between Hanna and Charlie is transcribed below:


*

*Charlie: I can not determine the number;

*Hanna: Neither can I, but I can tell if it is even or odd;

*Charlie: Now I know what the number is;

*Hanna: You know? Then I also know.


Suppose that students are honest and there is perfect logic in all the conclusions drawn. Determine the number thought by the teacher, justifying your answer.

The number is $2^6$.
Hanna knows that it has 2 a divider-- a number less than 100 with 6 dividers (7 if you count the number itself*) has to have 2 one of them.  And Charlie, figuring how she concluded that, sees 64 is the only one with 6 (96 also satisfies this part) divisors and digits adding up to the number given to him. 


*

*Edited to add the part in first parantheses.   

