Let $a_i,b_i,c_i (i=1,2)$ be integers such that the greatest common divisor $\gcd(a_i,b_i,c_i)=1$ for each $i$. Moreover suppose the triple $(a_1,b_1,c_1)$ is not a scale of $(a_2,b_2,c_2)$ i.e. there exist no $t\in \mathbb{Q}$ satisfying $a_1=t a_2,b_1=tb_2,c_1=tc_2$.
Question: Do there exist $m,n\in \mathbb{Z}$ such that $a_1+mb_1+nc_1$ is coprime with $a_2+mb_2+nc_2$?