Example 3 Given two numbers, the sum of the quotients obtained by dividing each by their greatest common divisor is 18, and their least common multiple is 975. Find these two positive integers.

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Let the two positive integers be $x, y$, and $d=(x, y)$, then
$$
x=d x_{1}, y=d y_{1},\left(x_{1}, y_{1}\right)=1 \text {. }
$$

Thus, $[x, y]=\frac{x y}{d}=d x_{1} y_{1}$.
According to the problem, we have
$$
\left\{\begin{array}{l}
x_{1}+y_{1}=18 \\
d x_{1} y_{1}=975
\end{array}\right.
$$

Clearly, $x_{1} \leqslant 17, y_{1} \leqslant 17$, and
$$
975=1 \cdot 3 \cdot 5 \cdot 5 \cdot 13 \text {. }
$$

Since $x_{1}, y_{1}$ must be divisors of 975, the only possible values are $x_{1}=5, y_{1}=13, d=15$.
Therefore, $x=75, y=195$.