$5 \cdot 17$. The sum of two numbers is 667, and the quotient of their least common multiple divided by their greatest common divisor is 120. Find these two numbers.
(Kyiv Mathematical Olympiad, 1954)
[Solution] Let the required numbers be $x$ and $y$, and $d=(x, y)$.
Then $x=d x_{1}, y=d y_{1},\left(x_{1}, y_{1}\right)=1$.
Thus, $[x, y]=\frac{x y}{d}=d x_{1} y_{1}$.
By the problem, we have
$$
\left\{\begin{array}{l}
d\left(x_{1}+y_{1}\right)=667 \\
\frac{d x_{1} y_{1}}{d}=120 .
\end{array}\right.
$$

Since $x_{1} y_{1}=120=2^{3} \cdot 3 \cdot 5$,
and $\quad 667=1 \cdot 23 \cdot 29$.
Therefore, from (1) and (2), we get

or
$$
x_{1}=8, y_{1}=15 \text {. }
$$
$$
x_{2}=24, y_{2}=5 \text {. }
$$

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