Determine whether there is an $x \in \mathbb{Z}$ such that $x^2 \equiv 2 \mod 15$.

Question

If we want for example to determine whether there is an $x \in \mathbb{Z}$ such that $x^2 \equiv 2 \mod 15$, it suffices to calculate $2^2=4,3^2=9,4^2=1,5^2=10,6^2=6,7^2=4$, because we have again $4$, we know that there is not such an $x$. Why is this true in particular? Why we know for sure that $8^2,9^2,...,14^2$ is again equal to one of previous squares?

Thanks in advance.

Answer 1

Modulo 15, $8=-7$ so $8^2=7^2$ etc.

Answer 2

You're argument is not right. You don't need to go through $8,\dotsc,14$ because they are the respective opposites (modulo $15$) of $7,\dotsc,1$, hence have the same squares.

Answer 3

$x^2\equiv2(mod 15) $ has solution if and only if $x^2\equiv2(mod 3)$ and $x^2\equiv2(mod 5)$ has solution. we note that $x^2\equiv2(mod 3)$ and $x^2\equiv2(mod 5)$ has no solution then $x^2\equiv2(mod 15) $ has no solution.