Given two sets of coprime integers: $\{a_1,\cdots,a_n\}$, $gcd(a_1,\cdots,a_n)=1$ and $\{b_1,\cdots,b_n\}$, $gcd(b_1,\cdots,b_n)=1$. Also given coprime integers $c_1$ and $c_2$. Is it always possible to find integer weights $\{k_1,\cdots,k_n\}$ such that $\sum_1^n k_ia_i$ is coprime with $c_1$ and $\sum_1^n k_ib_i$ is coprime with $c_2$ at the same time?
I feel this is possible, and the result can be generalized, but difficult to explain.