Solve the equation
 $mn =$ (gcd($m,n$))$^2$ + lcm($m, n$) 
in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.
**
   We have \( x = 2 \) and \( y = 4 \). This means:
   \[
   \gcd(m, n) = 2 \quad \text{and} \quad \text{lcm}(m, n) = 4
   \]
   Since \( \gcd(m, n) \cdot \text{lcm}(m, n) = mn \), we have:
   \[
   2 \cdot 4 = mn \implies mn = 8
   \]
   The pairs \((m, n)\) that satisfy \( \gcd(m, n) = 2 \) and \( mn = 8 \) are:
   \[
   (m, n) = (2, 4) \quad \text{or} \quad (m, n) = (4, 2)
   \]

Conclusion:
\[
\boxed{(m, n) = (2, 4) \text{ or } (4, 2)}
\]