Example 7 The sum of two numbers is 667, and the quotient of their least common multiple divided by their greatest common divisor equals 120. Find these two numbers.
Analysis: When encountering problems where the greatest common divisor (GCD) and least common multiple (LCM) have an equation relationship, generally set the two numbers in the problem as \(x, y\), \(d\) as the GCD of \(x, y\), \(x = d x_{1}, y = d y_{1}\), and the LCM as \(d x_{1} y_{1}\). Thus, we can establish equations involving \(d, x_{1}, y_{1}\), and solve for \(x, y\).

Solution: Let the numbers sought be \(x\) and \(y\), and \(d = (x, y)\),
then \(x = d x_{1}, y = d y_{1}, (x_{1}, y_{1}) = 1\),
thus \([x, y] = \frac{x y}{d} = d x_{1} y_{1}\),
by the problem, we have \(d (x_{1} + y_{1}) = 667\),
\[
\frac{d x_{1} y_{1}}{d} = 120 \text{, }
\]

Since \(x_{1} y_{1} = 120 = 2^{3} \cdot 3 \cdot 5\), and \(667 = 1 \cdot 23 \cdot 29\),
from (1) and (2), we get \(x_{1} = 8, y_{1} = 15\), or \(x_{2} = 24, y_{2} = 5\),
correspondingly, \(d_{1} = 29, d_{2} = 23\),
therefore, the numbers sought are two sets:
\[
x_{1} = 232, y_{1} = 435 \text{ or } x_{2} = 552, y_{2} = 115 \text{. }
\]