## Task A-4.5.

Determine (if it exists) the largest natural number that cannot be represented as the sum of some, not necessarily distinct, elements of the set $\{135,136,137, \ldots, 144\}$.
## First Solution.

We will say that a natural number is representable if it can be written as the sum of some, not necessarily distinct, elements of the set $\{135,136,137, \ldots, 144\}$. Let's denote the set of allowed addends $\{135,136,137, \ldots, 144\}$ as $S$ and define the sets

$$
S_{k}=\{135 k, 135 k+1, \ldots, 144 k\}
$$

We will prove that a number $n$ is representable as the sum of $k$ addends from $S$ if and only if $n \in S_{k}$.

If a number $n$ can be represented as the sum of $k$ elements of the set $S$, then it is clear that the number $n$ is in the set $S_{k}$ (since the $k$ addends in its sum are between 135 and 144).

Now let's prove the converse: every number in $S_{k}$ is representable as the sum of $k$ addends from $S$. Assume the contrary, that there are some numbers in $S_{k}$ that are not representable, and let $n_{0}$ be the smallest among them. Certainly, $n_{0} \neq 135 k$, since that number can be represented as the sum of $k$ repetitions of the addend 135. Therefore, $n_{0}>135 k$, so the number $n_{0}-1$ is representable. The number $n_{0}-1$ is less than $144 k$, so one of the addends in its representation is less than 144. By increasing that addend by 1 (while keeping all other addends the same), we obtain the sum of $k$ elements from $S$ which equals $n_{0}$, which is a contradiction to the non-representability of the number $n_{0}$. Therefore, all numbers between $135 k$ and $144 k$ are representable.

Now let's find the largest number that is not representable. If a natural number $m$ is not representable, then it does not belong to any set $S_{k}$. In particular, it lies between two consecutive sets $S_{k}$ and $S_{k+1}$, i.e., $144 k < m < 135(k+1)$, and this is possible if and only if

$$
144 k+2 \leqslant 135(k+1)
$$

from which we get $k \leqslant 14$. Therefore, for $k \geqslant 15$ there are no numbers that lie between $S_{k}$ and $S_{k+1}$, so the largest non-representable numbers are between $S_{14}$ and $S_{15}$.

The largest number between $S_{14}=\{1890, \ldots, 2016\}$ and $S_{15}=\{2025, \ldots, 2160\}$ is 2024, so this is also the largest non-representable number.