Problem
NT-B1-M06-P012 The Form \(x^2+2y^2\) Modulo \(5\)
#12
★★☆☆☆ Level 2 of 5
Prove that if \(5\mid x^2+2y^2\), then \(5\mid x\) and \(5\mid y\).
List the quadratic residues modulo \(5\): \(0,1,4\).
Squares modulo \(5\) are \(0,1,4\). Thus \(2y^2\) can have residues \(0,2,3\). The sum \(x^2+2y^2\) can be \(0\) modulo \(5\) only as \(0+0\). Therefore \(x^2\equiv0\) and \(y^2\equiv0\pmod5\), hence \(5\mid x\), \(5\mid y\).
The lemma prepares the stronger equation in the next problem.