Question Number 216842 by ArshadS last updated on 22/Feb/25
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{x},\:\mathrm{y}\:\:\mathrm{that}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\:\mathrm{system}\:\: \\ $$$$\mathrm{xy}\:+\:\mathrm{x}\:+\:\mathrm{y}=\mathrm{71}\: \\ $$$$\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} =\mathrm{880} \\ $$
Answered by Frix last updated on 22/Feb/25
$${xy}+\left({x}+{y}\right)=\mathrm{71} \\ $$$${xy}\left({x}+{y}\right)=\mathrm{880} \\ $$$${xy}=\mathrm{71}−\left({x}+{y}\right) \\ $$$$\left(\mathrm{71}−\left({x}+{y}\right)\right)\left({x}+{y}\right)=\mathrm{880} \\ $$$$\mathrm{71}\left({x}+{y}\right)−\left({x}+{y}\right)^{\mathrm{2}} =\mathrm{880} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{71}\left({x}+{y}\right)+\mathrm{880}=\mathrm{0} \\ $$$$\left({x}+{y}\right)=\frac{\mathrm{71}}{\mathrm{2}}\pm\frac{\mathrm{39}}{\mathrm{2}} \\ $$$$\left({x}+{y}\right)=\mathrm{55}\vee\left({x}+{y}\right)=\mathrm{16} \\ $$$$\Rightarrow\:\left({x}+{y}\right)=\mathrm{55}\wedge{xy}=\mathrm{16}\:\mathrm{impossible} \\ $$$$\vee\:\:\left({x}+{y}\right)=\mathrm{16}\wedge{xy}=\mathrm{55} \\ $$$$\Rightarrow\:{x}=\mathrm{5}\wedge{y}=\mathrm{11}\:\vee\:{x}=\mathrm{11}\wedge{y}=\mathrm{5} \\ $$
Commented by ArshadS last updated on 22/Feb/25
$$\mathbb{N}\mathrm{ice}\:\mathrm{sir}! \\ $$