$35 \cdot 22$ A supermarket has 128 boxes of apples. Each box contains between 120 and 144 apples. Boxes with the same number of apples are called a group. What is the minimum number $n$ of boxes in the largest group?
(A) 4 .
(B) 5 .
(C) 6 .
(D) 24 .
(E) 25 .
(27th American High School Mathematics Examination, 1976)
[Solution] The number of apples that can be packed in 1 box is $120, 121, \cdots, 144$, which gives 25 different ways. If each way is used to pack 5 boxes, then at most $25 \cdot 5 = 125$ boxes.
Since there are 128 boxes now, we have $n \geqslant 6$.
Because there is the following possibility: boxes containing $k$ apples $(k=120,121,122)$ each have 6 boxes, and boxes containing $l$ apples $(123 \leqslant l \leqslant 144)$ each have 5 boxes. This way, $n \leqslant 6$.
Therefore, $n=6$.
So the answer is $(C)$.
Note: If the total number of boxes is 126, it is a problem from the 2nd Tianjin "Chinese Youth Forest" Mathematics Invitational Competition (1987).
And 126 is the minimum number of boxes to ensure the conclusion of the proposition holds.