A pair $(a,b)$ of positive integers is good if $\gcd(a,b)=1$ and for each pair of sets $A,B$ of positive integers such that $A,B$ are, respectively, complete residues system modulo $a,b$, there are $x \in A, y \in B$ such that $\gcd(x+y,ab)=1$. For each pair of positive integers $a,k$, let $f(N)$ the number of $b \leq N$ such $b$ has $k$ distinct prime factors and $(a,b)$ is good. Prove that \[\liminf_{n \to \infty} f(n)/\frac{n}{(\log n)^k}\ge e^{k}\]