Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the [i]Primle[/i]. For each guess, a digit is highlighted blue if it is in the [i]Primle[/i], but not in the correct place. A digit is highlighted orange if it is in the [i]Primle[/i] and is in the correct place. Finally, a digit is left unhighlighted if it is not in the [i]Primle[/i]. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the [i]Primle[/i]?
$$\begin{array}{c} 
            \boxed{1} \,\, \boxed{3}  \\[\smallskipamount]
            \boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7}
       \end{array}$$

[i]Proposed by Andrew Wu and Jason Wang[/i]
1. We are given that Charlotte guesses the numbers $13$ and $47$ and receives the following feedback:
   \[
   \begin{array}{c} 
       \boxed{1} \,\, \boxed{3}  [\smallskipamount]
       \boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7}
   \end{array}
   \]
   This feedback indicates that:
   - The digits $1$, $3$, and $4$ are not in the *Primle*.
   - The digit $7$ is in the *Primle* but not in the correct place.

2. Since $7$ is in the *Primle* but not in the units place (as indicated by the blue highlight), it must be in the tens place. Therefore, the *Primle* is a two-digit number in the $70$s.

3. The prime numbers in the $70$s are $71$, $73$, and $79$.

4. We can eliminate $71$ and $73$ because:
   - If the *Primle* were $71$, the digit $1$ would be in the *Primle*, but the feedback from the guess $13$ indicates that $1$ is not in the *Primle*.
   - If the *Primle* were $73$, the digit $3$ would be in the *Primle*, but the feedback from the guess $13$ indicates that $3$ is not in the *Primle*.

5. Therefore, the only remaining possibility is $79$.