I am studying the theorem that the factorization into irreducibles in $\mathbb{Z}(\sqrt{15})$ is not unique.
I am stuck in proving that the equations $a^2\equiv{2,5,13,10} \pmod{15}$ have no solution in $\mathbb{Z}$.
I saw how a linear congruence equation can be solved but I do not know how to solve this equations in order to prove that each of them do not have solution.
Would you help me, please? Thank you in advance.
I recommend you use the CRT:
$$\mathbb{Z}_{15} \;\;\;\;\;\;\rightarrow \;\;\;\;\mathbb{Z}_5\;\;\;\;\;\;\;\;\;\; \times \;\;\;\;\;\;\;\;\mathbb{Z}_3 \\ x^2 \;\;\rightarrow \;\;\;\; (x^2 \bmod 3)\;\;\;\;\times \;\;\;(x^2 \bmod 5) \\ $$ and solve the quadratic modulo equation in each of the smaller field.If you can find a solution in each of the smaller field, then you can find a solution in $\mathbb{Z}_{15}$.