For which odd primes $p ≠ 5$ is 10 a qudratic residue modulo $p$?

Question

For which odd primes $p ≠ 5$ is 10 a quadratic residue modulo $p$?

Saw a similar example using 5 and 15 and did my best to learn from those but still having a hard time grasping how to complete this problem. Any help is appreciated.

Answer 1

$10$ is a quadratic residue $\!\!\pmod{p}$ if both $2$ and $5$ are quadratic residues or neither $2$ or $5$ are quadratic residue. By quadratic reciprocity, $2$ is a quadratic residue $\!\!\pmod{p}$ iff $p\equiv\pm 1\pmod{8}$ and $5$ is a quadratic residue iff $p\equiv \pm 1\pmod{10}$. Now we may use the Chinese remainder thereom to state that $10$ is a quadratic residue $\!\!\pmod{p}$ iff $p\in \{1,3,9,13,27,31,37,39\}\pmod{40}$.