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.

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- May 11th 2009, 01:44 PMcathwelchLegendre 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. - May 11th 2009, 02:56 PMPaulRS
Prove it by contradiction, assume none of them were a quadratic residue, then $\displaystyle

\left( {\tfrac{a}

{p}} \right) = \left( {\tfrac{b}

{p}} \right) = \left( {\tfrac{{ab}}

{p}} \right) = - 1

$ however remember that $\displaystyle

\left( {\tfrac{a}

{p}} \right) \cdot \left( {\tfrac{b}

{p}} \right) = \left( {\tfrac{{ab}}

{p}} \right)

$ (Wink)