$10 \cdot 87$ Four three-digit numbers with the same first digit are distinct, and have the property: the sum of these numbers is divisible by any three of them. Find these four numbers.
(3rd All-Soviet Union Mathematical Olympiad, 1969)
[Solution] Let the four numbers we are looking for be $x_{1}, x_{2}, x_{3}, x_{4}$, and
$$
S=x_{1}+x_{2}+x_{3}+x_{4} \text {. }
$$

Let $a$ be the leading digit of $x_{1}, x_{2}, x_{3}, x_{4}$, then
$$
100 a \leqslant x_{i} < 1000 a, \quad i=1,2,3,4.
$$
If $a \geqslant 2$, then $100 a \geqslant 200$, and thus $x_{i} \geqslant 200$. Therefore, $S \geqslant 4 \times 200 = 800$. However, the sum of the four numbers is $540$, which is less than $800$. Therefore, such four numbers are impossible.
Thus, the four numbers we are looking for are $108,135,180,117$.