14. In a school there are 1000 students, numbered 1 to 1000 . A group of 500 students is said to be a 'good group' if there exists a student in the group whose number divides the number of another student in the group, and is said to be a 'bad group' if otherwise. For instance, the 500 students with numbers 1 to 500 form a 'good group' because 13 divides 26 and both students 13 and 26 are in the group. A 'good student' is a student who does not belong to any 'bad group'. Find, among the 'good students', the one with the largest student number.
(2 marks)
某校有 1000 名學生, 分別編號為 1 至 1000 。當中任何 500 名學生中, 若有一人的編號整除另一人的編號, 則這 500 名學生稱為「好群組」, 否則稱為「壞群組」。例如：編號 1 至 500 的學生組成一個「好群組」，因為編號 13 和 26 的學生均在這群組中, 而且 13 整除 26 。若某名學生不屬於任何「壞群組」, 則稱為「好學生」。在所有「好學生」中, 求編號最大的一個。
14. 333
14. We first show that student 333 is a 'good student'. Divide the 997 students (all except students 333, 666 and 999) into the following 498 sets:
$$
\{1,2,4, \ldots\},\{3,6,12, \ldots\},\{5,10,20, \ldots\}, \ldots,\{331,662\},\{335,670\}, \ldots,\{995\},\{997\}
$$

Now consider a group of 500 students containing student 333 . If either student 666 or 999 is in the group, then it is a 'good group'. Otherwise, the other 499 students of the group come from the above 498 sets, and the pigeon-hole principle asserts that two of them come from the same set. For any two students coming from the same set, one must have a student number which divides the other's. It follows that this must be a 'good group', and hence student 333 is a 'good student'.

Finally we show that students 334 to 1000 are not 'good students'. Clearly, students 501 to 1000 form a 'bad group', and hence students 501 to 1000 are not 'good students'. Now consider the group formed by the following 500 students:
$$
334,335, \ldots, 667,669,671,673,997,999 .
$$

This group consists of students 334 to 667 and all odd-numbered students from 669 to 999 . It is easy to see that these students form a 'bad group', because for $334 \leq n \leq 500$, student $n$ is in the group but student $2 n$ is not, and $3 n$ already exceeds 1000 . For $n \geq 501,2 n$ already exceeds 1000. Hence we can't find two students for which the student number of one divides that of another. It follows that this is a 'bad group', and hence students 334 to 500 are not 'good students'. Consequently, the answer is 333 .