2016 APMO 第 2 题

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.

Answer: The answer is $n=2^{101}-1$.

如果一个正整数可以用以下形式表示，则称为花式

$$2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}$$

其中 $a_{1}, a_{2}, \ldots, a_{100}$ 是不一定不同的非负整数。
找到最小的正整数 $n$，使得 $n$ 的倍数都不是奇数。
答案：答案是$n=2^{101}-1$。