The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1. **Determine the total number of days in the year 2020:**
   Since 2020 is a leap year (divisible by 4 and not a century year unless divisible by 400), it has 366 days.

2. **Calculate the mean number of days per month:**
   The mean number of days per month is given by the total number of days divided by the number of months.
   \[
   \text{Mean number of days per month} = \frac{366 \text{ days}}{12 \text{ months}}
   \]

3. **Simplify the fraction:**
   \[
   \frac{366}{12} = \frac{366 \div 6}{12 \div 6} = \frac{61}{2}
   \]
   Here, 366 and 12 are both divisible by 6, so we simplify the fraction by dividing both the numerator and the denominator by 6.

4. **Identify the relatively prime integers \( m \) and \( n \):**
   The fraction \(\frac{61}{2}\) is already in its simplest form, where \( m = 61 \) and \( n = 2 \).

5. **Calculate \( m + n \):**
   \[
   m + n = 61 + 2 = 63
   \]

The final answer is \(\boxed{63}\).