1. ## Legendre Symbol

Question:
If p is an odd prime and gcd (ab, p)=1, prove that at least one of a, b, or ab is a quadratic residue of p.

2. Prove it by contradiction, assume none of them were a quadratic residue, then $
\left( {\tfrac{a}
{p}} \right) = \left( {\tfrac{b}
{p}} \right) = \left( {\tfrac{{ab}}
{p}} \right) = - 1
$
however remember that $
\left( {\tfrac{a}
{p}} \right) \cdot \left( {\tfrac{b}
{p}} \right) = \left( {\tfrac{{ab}}
{p}} \right)
$