Question Number 211680 by RojaTaniya last updated on 16/Sep/24
$$\:{x},\:{y}\:{are}\:{positive}\:{integer}\:{such}\: \\ $$$$\:\:{that},\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{xy}=\mathrm{911}.\:\left({x},{y}\right)=? \\ $$
Commented by AlagaIbile last updated on 16/Sep/24
$$\:{That}'{s}\:{a}\:{symmetric}\:{Function} \\ $$$$\:\:\left({x}\:+\:{y}\right)^{\mathrm{3}} \:−\:\mathrm{3}{xy}\left({x}\:+\:{y}\right)\:+\:{xy}\:=\:\mathrm{911} \\ $$$$\:\:\boldsymbol{{a}}^{\mathrm{3}} \:−\:\mathrm{3}\boldsymbol{{ba}}\:+\:\boldsymbol{{b}}\:=\:\mathrm{911} \\ $$$$\:\:\boldsymbol{{b}}\:=\:\frac{\boldsymbol{{a}}^{\mathrm{3}} \:−\:\mathrm{911}}{\mathrm{3}\boldsymbol{{a}}\:−\:\mathrm{1}} \\ $$$$\:{Trying}\:\boldsymbol{{a}}\:=\:\mathrm{10},\:\mathrm{11},… \\ $$$$\:\:\boldsymbol{{b}}\:=\:\frac{\mathrm{15}^{\mathrm{3}} \:−\:\mathrm{911}}{\mathrm{3}\left(\mathrm{15}\right)−\:\mathrm{1}}\:=\:\mathrm{56} \\ $$$$\:\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:=\:\mathrm{15}\:,\:\boldsymbol{{xy}}\:=\:\mathrm{56} \\ $$$$\:\therefore\:\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)\:=\:\left(\mathrm{8},\mathrm{7}\right)\:,\:\left(\mathrm{7},\mathrm{8}\right) \\ $$
Answered by A5T last updated on 16/Sep/24
$${WLOG},\:{let}\:{x}\geqslant{y}\Rightarrow\mathrm{911}={x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{xy}\geqslant\mathrm{2}{y}^{\mathrm{3}} +{y}^{\mathrm{2}} \\ $$$$\Rightarrow{y}\leqslant\mathrm{7}.\:{Observe}\:{that}\:{x}\leqslant\mathrm{9},\:{because}\:\mathrm{10}^{\mathrm{3}} >\mathrm{911} \\ $$$${Checking}\Rightarrow{y}=\mathrm{7},{x}=\mathrm{8} \\ $$$$\Rightarrow\left({x},{y}\right)=\left(\mathrm{8},\mathrm{7}\right)\:{upto}\:{permutation}. \\ $$
Commented by RojaTaniya last updated on 16/Sep/24
$${Sir},\:{perfect}\:{idea}.\:{Thanks}. \\ $$