Question 99
Let $n$ be a positive integer and $u_{1}, u_{2}, \ldots, u_{n}$ be positive integers not larger than $2^{k}$, for some integer $k \geq 3$. A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_{1}, a_{2}, \ldots, a_{n}$ such that
$$
t = a_{1} u_{1} + a_{2} u_{2} + \cdots + a_{n} u_{n}
$$
Prove that if a non-negative integer $t$ has a representation, then it also has a representation where less than $2k$ of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ are non-zero.
$$
t = a_{1} u_{1} + a_{2} u_{2} + \cdots + a_{n} u_{n}
$$
Prove that if a non-negative integer $t$ has a representation, then it also has a representation where less than $2k$ of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ are non-zero.