Quadratic reciprocity - legendre symbol $\neq$ jacobi symbol

Question

I want to calculate wether $\exists x : x^2 \equiv 123 \mod 11\cdot 13$ or not. I do know that in terms of the legendre symbol follows that $\neg(\exists x: x^2 \equiv 123 \mod 11)$ and $\neg(\exists x: x^2 \equiv 123 \mod 13)$. How can i deduce from that, that $\neg(\exists x: x^2 \equiv 123 \mod 11 \cdot 13)$ ? The intention is that the values of the legende-symbol and the jacobi-symbol do not have to be equal.

Answer 1

Assume there exists $x$ such that $x^2 \equiv 123 \pmod{11\cdot 13}$. Then, by definition, $$x^2 = 123 + 11\cdot 13\cdot k = 123 + 11(13k)$$ which implies that $x^2 \equiv 123 \pmod{11}$. Contradiction.