Consider some system of congruences $$ \left\{ \begin{aligned} x &\equiv r_1 \pmod {p_1} \\ x &\equiv r_2 \pmod {p_2} \\ &\;\;\vdots \\ x &\equiv r_n \pmod{p_n} \end{aligned} \right. $$
Where $p_1, p_2, \dots, p_n$ are the first $n$ prime numbers.
Suppose that the system has a solution, and thus the general solution can be found using the Chinese remainder Theorem. My question is: is there some method to systematically find the minimum integer solution to the system? If not, is there at least some method to find some lower bound, and how sharp?
Thanks!