7. There are $n$ different positive integers, each one not greater than 2013 , with the property that the sum of any three of them is divisible by 39 . Find the greatest value of $n$.
(1 mark)
現有 $n$ 個互不相同且每一個都不大於 2013 的正整數, 且當中任意三個之和均可被 39 整除。求 $n$ 的最大值。
7. 52
7. Let $a, b, c, d$ be among the positive integers. Then both $a+c+d$ and $b+c+d$ are divisible by 39 , and so is their difference $a-b$. It follows that any two of the integers are congruent modulo 39. Since $2013=39 \times 51+24$, at most 52 integers can be chosen. Indeed, if we choose the 52 integers in the set $\{13,52,91, \ldots, 2002\}$, then the sum of any three is divisible by 39 (as each one is congruent to 13 modulo 39). It follows that the answer is 52 .