Let $a_1,a_2,b_1,b_2,c_1,c_2,d_1$ be known positive integers. Let $x$ be an unknown satisfying $$ 4(a_1d_1+1) \mid (a_1x+d_1a_2)^2$$ $$ 4(b_1d_1+1) \mid (b_1x+d_1b_2)^2$$ $$ 4(c_1d_1+1) \mid (c_1x+d_1c_2)^2.$$
Where, the integers $a_1,b_1,c_1$ and $d_1$ are distinct. I want to know whether solution exist, if so is it unique?
I think if solution exists, it can not be unique. Suppose $x=d$ is a solution, then so is $d+4k(a_1d_1+1)(b_1d_1+1)(c_1d_1+1)$ for any $k \in \mathbb{Z}.$
In my view, solution need not always exist.
For example if $a~|~bx+c$ then $bx+c = ka$ which means $x = \frac{ka-c}{b}$ which need not always be an integer.
Thus I am curious to know under what condition does the solution exists for the above system?