Let $u$ be the greatest common divisor of the positive integers $x$ and $y$, and $v$ be the least common multiple of $x$ and $y$. Solve the following system of equations:

$$
\begin{aligned}
x \cdot y \cdot u \cdot v & =3600 \\
u+v & =32
\end{aligned}
$$
The product of the greatest common divisor and the least common multiple of two numbers is equal to the product of the numbers. Therefore, it follows from the first equation that $u \cdot v=60$. Given the condition $u \leq v$, the system of equations

$$
\begin{aligned}
u v & =60 \\
u+v & =32
\end{aligned}
$$

has a unique solution: $u=2, v=30$. Since the least common multiple $v=30=2 \cdot 3 \cdot 5$, in its prime factorization, every prime number appears to the first power, hence the solution to the problem is:

$$
\begin{gathered}
x=6 \quad \text { and } \quad y=10 \quad \text { or } \\
x=10 \quad \text { and } \quad y=6
\end{gathered}
$$