Question Number 216911 by ArshadS last updated on 24/Feb/25
$${Find}\:{all}\:{positive}\:{integer}\:\mathrm{x},\mathrm{y}\:{such}\:{that} \\ $$$$\mathrm{x}^{\mathrm{2}} +\:\mathrm{y}^{\mathrm{2}} +\:\mathrm{xy}\:=\:\mathrm{169} \\ $$
Answered by A5T last updated on 25/Feb/25
$$\mathrm{WLOG},\:\mathrm{let}\:\mathrm{x}\geqslant\mathrm{y} \\ $$$$\Rightarrow\mathrm{169}=\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{xy}\geqslant\mathrm{3y}^{\mathrm{2}} \Rightarrow\mathrm{y}^{\mathrm{2}} \leqslant\frac{\mathrm{169}}{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{y}\leqslant\mathrm{7} \\ $$$$\mathrm{Checking}\Rightarrow\left(\mathrm{x},\mathrm{y}\right)=\left(\mathrm{8},\mathrm{7}\right) \\ $$
Commented by ArshadS last updated on 25/Feb/25
$$\mathrm{Nice}\:\mathrm{sir}! \\ $$