APMO 2016 Problem 2

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.