Question 21
A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}} + 2^{a_{2}} + \cdots + 2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
$$
2^{a_{1}} + 2^{a_{2}} + \cdots + 2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.