Problem #WSP - 000088

Problems Combinatorics Number Theory

Problem

There is a secret gathering of a group of \(n\) aliens in a very dark room. You cannot see anyone in the room, but you hear the following questions.

  1. “Is at least one of us a Goop?"

  2. “Is the number of Goops amongst us an even number?"

  3. “Is the number of Goops amongst us a multiple of 3?"

  4. \(\dots\)

  5. “Is the number of Goops amongst us a multiple of \(n\)?"

What are all the possible values of \(n\) such that this gathering can happen? Note that each of the \(n\) aliens have asked exactly one question.