2. 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.
2. Let the required numbers be $x$ and $y$, and let $d=(x, y)$, then
$$
x=d x_{1}, y=d y_{1},\left(x_{1}, y_{1}\right)=1 \text {. }
$$

Thus, $[x, y]=\frac{x y}{d}=d x_{1} y_{1}$.
By the problem, we have
$$
\left\{\begin{array}{l}
d\left(x_{1}+y_{1}\right)=667, \\
\frac{d x_{1} y_{1}}{d}=120 .
\end{array}\right.
$$
(1).
From $x_{1} y_{1}=120=2^{3} \cdot 3 \cdot 5$,
and $667=1 \cdot 23 \cdot 29$,
therefore, from (1) and (2) we get
$$
x_{1}=8, y_{1}=15 \text {, or } x_{2}=24, y_{2}=5 \text {. }
$$

The corresponding $d_{1}=29, d_{2}=23$.
Thus, the required numbers are two sets:
$$
x_{1}=232, y_{1}=435 \text {, or } x_{2}=552, y_{2}=115 \text {. }
$$