When is "mod $n$" a congruence relation on the lattice $(\Bbb N,\gcd,\text{lcm})$? For which $n\in \Bbb N$,
$$a\equiv b,a'\equiv b'\quad \text{implies} \quad \gcd(a,a')\equiv \gcd(b,b'), 
\text{lcm}(a,a')\equiv \text{lcm}(b,b')$$
all mod $n$.

For $n=2$ it is true.


This is trivially true when $n=1$, and easily verified to be true when $n=2$. Consider $n \geq 3$. Consider 
\begin{equation}
1 \equiv (n-1)^2 \pmod{n}, n-1 \equiv n-1 \pmod{n} \\
\gcd(1, n-1)=1 \not \equiv -1 \equiv \gcd((n-1)^2, n-1) \pmod{n}
\end{equation}
Therefore it is only true for $n=1, 2$.
