I am self-studying number theory and came across the question asked by another user in this post here. Specifically part (b).
My initial solution is to use the fact that the Legendre Symbol is completely multiplicative to split $(\frac{-5}{p})$ = $(\frac{-1}{p})*(\frac{5}{p})$, then follow the same steps to compute $(\frac{5}{p})$. Since $(\frac{-1}{p})$ is equal to 1 if $p \equiv 1\bmod\ 4$ and $-1$ if $p \equiv 3\bmod\ 4$, my idea was to use the CRT to find solutions that satisfy the congruence system. However, I am not sure this is the right next step. Any hint as to how I should go about solving this question?