Question 44
Given a positive integer $n$, we say two sets $A$ and $B$ of integers are congruent modulo $n$ if and only if the residues of elements of $A$ modulo $n$ are the same as the residues of elements of $B$ modulo $n$, and we write $A \equiv B \pmod{n}$. For example,
$$
\{3, 5, 8\} \equiv \{-4, 0, 9, 15\} \pmod{3}
$$
We say that two sets $A$ and $B$ of integers are translatable modulo $n$ if there is a polynomial $f(x)$ with integer coefficients such that either $f(A) \equiv B \pmod{n}$ or $f(B) \equiv A \pmod{n}$. Here, by $f(X)$ for a subset $X$ of integers we mean $f(X) = \{f(x) : x \in X\}$.
Find all positive integers $n \gt 4$ such that every two non-empty subsets of $\{0, 1, \dots, n-1\}$ are translatable modulo $n$.
$$
\{3, 5, 8\} \equiv \{-4, 0, 9, 15\} \pmod{3}
$$
We say that two sets $A$ and $B$ of integers are translatable modulo $n$ if there is a polynomial $f(x)$ with integer coefficients such that either $f(A) \equiv B \pmod{n}$ or $f(B) \equiv A \pmod{n}$. Here, by $f(X)$ for a subset $X$ of integers we mean $f(X) = \{f(x) : x \in X\}$.
Find all positive integers $n \gt 4$ such that every two non-empty subsets of $\{0, 1, \dots, n-1\}$ are translatable modulo $n$.