Andy knows neither of them know the number, which means the ones digit could not be $7.$  Then using elimination, Fiona knows it, which means the tens digit could be $3$ or $6.$ But no matter what it is, the ones digit will always be $1.$ 
Mr. John plays a game with his two smart students.  He says: "I have a $2-$digit prime number. But I tell Andy only the ones digit and tell Fiona only the tens digit."  A few moments later, Andy says: "I don\textquotesingle t know what the number is, neither do you."  Fiona says: "I didn\textquotesingle t know what the number was before, but now I know it."  What is the ones digit of the prime number? 
Options:
- $$1$$ 
- $$2$$ 
- $$3$$ 
- $$5$$ 
- $$7$$ 