Determine all positive integers $n \ge 2$ that satisfy the following condition: for $a,b$ relatively prime to $n$ we have $a \equiv b \pmod n$ if and only if $ab \equiv 1 \pmod n$.
The condition is equivalent to the condition that $a \equiv b \pmod n \iff a^2 \equiv 1 \pmod n$. Thus we're looking for $n$'s that satisfy $a^2 \equiv 1 \pmod n$.
Now by the Chinese remainder theorem we can split this congruence using the prime factorization of $n$ and I think we will get that if $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$, then $$\begin{align} a^2 &\equiv 1 \pmod{p_1^{\alpha_1}} \\ &\vdots \\a^2 & \equiv 1 \pmod{p_k^{\alpha_k}}\end{align}$$
the question is now how I can use this information to get some equivalent statement from which I can derive the information for $n$?
