Question

Solve in positive integers:
$5x^2-10xy+7y^2=77$.

Answer 1

$5(x^2-2xy+y^2)+2y^2=77$

$5(x-y{)}^2+2y^2=77$

$77=45+32=5\left(9\right)+2\left(16\right)$

By comparision $(x-y{)}^2=9;y^2=16$

$=>y=4$ Since $y\ge 0$

$y=4$

case1: $x-y=3=>x=7$

case2:$x-y=-3=>x=1$

$77=5+72=5\left(1\right)+2\left(36\right)$

By comparision $(x-y{)}^2=1;y^2=36$

$=>y=6$ Since $y\ge 0$

$y=6$

case1: $x-y=1=>x=7$

case2:$x-y=-1=>x=5$

Hence $(x,y)=(7,4);(1,4);(7,6);(5,6)$