7. Find the largest integer which cannot be expressed as sum of some of the numbers $135,136,137, \ldots, 144$ (each number may occur many times in the sum or not at all).
(1 mark)
某整數不能寫成 135、136、‥144 當中的某些數之和（在和式中每個數出現的次數不限, 亦可以不出現）。求該整數的最大可能值。
7. 2024
7. Let $n$ be the integer. Note that there must be no integer $k$ for which $\frac{n}{144} \leq k \leq \frac{n}{135}$, for if such integer exists, we must be able to write $n$ as the sum of $k$ numbers, each between 135 and 144 . For instance, if $n=2500$, then $\frac{n}{144} \approx 17.4$ and $\frac{n}{135} \approx 18.5$. We may then take $k=18$. Since $135 \times 18=2430$ and $144 \times 18=2592$, to form 2500 we may start by using 135 only (and we are short by 70), and then change 14 of the 135 to 140 , thus getting $2500=135 \times 4+140 \times 14$.
If $\frac{n}{135}-\frac{n}{144} \geq 1$, then the above-mentioned integer $k$ must exist. Hence $\frac{n}{135}-\frac{n}{144}<1$ and so $9 n<135 \times 144$, or $n<135 \times 16=2160$. When $n$ is slightly less than 2160 , we can still take $k=15$ until $n$ is less than $135 \times 15=2025$.
Indeed, the answer is 2024. This is because with 14 summands, the value of $n$ is at most $144 \times 14=2016$; with 15 summands, the value of $n$ is at least $135 \times 15=2025$. It follows that 2024 cannot be expressed in the form described in the question.