Determine all possible values of $m+n$, where $m$ and $n$ are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]
1. Let \( m = d m' \) and \( n = d n' \) where \( \gcd(m, n) = d \). We know that \( \operatorname{lcm}(m, n) = \frac{m \cdot n}{\gcd(m, n)} \). Therefore, 
   \[
   \operatorname{lcm}(m, n) = \frac{d m' \cdot d n'}{d} = d m' n'.
   \]
   Given that \( \operatorname{lcm}(m, n) - \gcd(m, n) = 103 \), we substitute to get:
   \[
   d m' n' - d = 103.
   \]
   Factoring out \( d \) from the left-hand side, we have:
   \[
   d (m' n' - 1) = 103.
   \]
   Since 103 is a prime number, the possible values for \( d \) are 1 and 103.

2. **Case 1: \( d = 103 \)**
   \[
   103 (m' n' - 1) = 103 \implies m' n' - 1 = 1 \implies m' n' = 2.
   \]
   The pairs \((m', n')\) that satisfy \( m' n' = 2 \) and are relatively prime are \((1, 2)\) and \((2, 1)\). Thus, the corresponding values of \( m \) and \( n \) are:
   \[
   m = 103 \cdot 1, \quad n = 103 \cdot 2 \quad \text{or} \quad m = 103 \cdot 2, \quad n = 103 \cdot 1.
   \]
   In both cases, \( m + n = 103 + 206 = 309 \).

3. **Case 2: \( d = 1 \)**
   \[
   1 (m' n' - 1) = 103 \implies m' n' - 1 = 103 \implies m' n' = 104.
   \]
   The pairs \((m', n')\) that satisfy \( m' n' = 104 \) and are relatively prime are \((1, 104)\), \((104, 1)\), \((8, 13)\), and \((13, 8)\). Thus, the corresponding values of \( m \) and \( n \) are:
   \[
   m = 1 \cdot 1, \quad n = 1 \cdot 104 \quad \text{or} \quad m = 1 \cdot 104, \quad n = 1 \cdot 1,
   \]
   \[
   m = 1 \cdot 8, \quad n = 1 \cdot 13 \quad \text{or} \quad m = 1 \cdot 13, \quad n = 1 \cdot 8.
   \]
   Therefore, the possible values of \( m + n \) are:
   \[
   1 + 104 = 105 \quad \text{and} \quad 8 + 13 = 21.
   \]

Combining all the possible values from both cases, we get:
\[
\boxed{21, 105, 309}.
\]