A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit positive integers that are [i]literally 1434[/i].

[i]Proposed by Evin Liang[/i]

[hide=Solution]
[i]Solution.[/i] $\boxed{67384}$
The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$, $b$ is $4$ or $9$, $c$ is $3$ or $8$, and $d$ is $4$ or $9$. There are $16$ such numbers and the average is $\dfrac{8423}{2}$, so the total in this case is $\boxed{67384}$.
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To solve this problem, we need to identify all four-digit numbers \( n \) such that when each digit is replaced by its remainder when divided by 5, the result is 1434. 

1. **Identify the possible digits:**
   - The first digit \( a \) must satisfy \( a \mod 5 = 1 \). Therefore, \( a \) can be 1 or 6.
   - The second digit \( b \) must satisfy \( b \mod 5 = 4 \). Therefore, \( b \) can be 4 or 9.
   - The third digit \( c \) must satisfy \( c \mod 5 = 3 \). Therefore, \( c \) can be 3 or 8.
   - The fourth digit \( d \) must satisfy \( d \mod 5 = 4 \). Therefore, \( d \) can be 4 or 9.

2. **List all possible numbers:**
   - The possible values for \( a \) are 1 and 6.
   - The possible values for \( b \) are 4 and 9.
   - The possible values for \( c \) are 3 and 8.
   - The possible values for \( d \) are 4 and 9.

   Therefore, the possible numbers are:
   \[
   \begin{aligned}
   &1434, 1439, 1484, 1489, \\
   &1934, 1939, 1984, 1989, \\
   &6434, 6439, 6484, 6489, \\
   &6934, 6939, 6984, 6989.
   \end{aligned}
   \]

3. **Calculate the sum of all these numbers:**
   - Sum of numbers starting with 1:
     \[
     1434 + 1439 + 1484 + 1489 + 1934 + 1939 + 1984 + 1989
     \]
   - Sum of numbers starting with 6:
     \[
     6434 + 6439 + 6484 + 6489 + 6934 + 6939 + 6984 + 6989
     \]

4. **Sum of numbers starting with 1:**
   \[
   \begin{aligned}
   &1434 + 1439 + 1484 + 1489 + 1934 + 1939 + 1984 + 1989 \\
   &= (1434 + 1484 + 1934 + 1984) + (1439 + 1489 + 1939 + 1989) \\
   &= 6836 + 6856 \\
   &= 13692
   \end{aligned}
   \]

5. **Sum of numbers starting with 6:**
   \[
   \begin{aligned}
   &6434 + 6439 + 6484 + 6489 + 6934 + 6939 + 6984 + 6989 \\
   &= (6434 + 6484 + 6934 + 6984) + (6439 + 6489 + 6939 + 6989) \\
   &= 26836 + 26856 \\
   &= 53692
   \end{aligned}
   \]

6. **Total sum:**
   \[
   13692 + 53692 = 67384
   \]

The final answer is \(\boxed{67384}\)