Q:

True or false: There is no square $6$ mod $7$.

True or false: There is no square $6$ mod $7$. If you find an example, then you are finish.  If you cannot find an example, then prove that the below statement is not true. 
$$ x^2 \equiv 6\mod 7$$
When I try some examples I get $0,1,2,4 \mod 7$ so I would have to prove that there is no square $6 \mod 7$ but I am having a hard time.  Any ideas? 

A:

Suppose $x^2\equiv 6$ then $x^4\equiv 36 \equiv 1$, and clearly $x$ is not a multiple of $7$, so little Fermat tells us that $x^6\equiv 1$ but then $x^6=x^2\cdot x^4\equiv 6\times 1\equiv 6$ is a contradiction. 
This also shows by easy generalisation that the congruence $x^2\equiv -1 \bmod p$ cannot be solved for any prime $p\equiv 3 \bmod 4$

A:

Hint: prove for $x=1,2,3,4,5,6$

