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Question Number 69111 by Fawole last updated on 20/Sep/19

Commented by Rasheed.Sindhi last updated on 20/Sep/19

(2,11),(11,2),(5,10) & (10,5).

$$\left(\mathrm{2},\mathrm{11}\right),\left(\mathrm{11},\mathrm{2}\right),\left(\mathrm{5},\mathrm{10}\right)\:\&\:\left(\mathrm{10},\mathrm{5}\right). \\ $$

Commented by Fawole last updated on 20/Sep/19

Show workings pleasesir

$${Show}\:{workings}\:{pleasesir} \\ $$

Answered by Rasheed.Sindhi last updated on 21/Sep/19

m^6 +n^6 +375m^2 n^2 =1953125 ⇒m^6 +n^6 +3m^2 n^2 (125)=(125)^3 Compare it with: (m^2 )^3 +(n^2 )^3 +3m^2 n^2 (m^2 +n^2 ) =(m^2 +n^2 )^3 ⇒m^2 +n^2 =125 m=(√(125−n^2 )) Only for n=2,11,5,10 m∈Z^+ Hence The solutions are (m,n)=(2,11),(11,2),(5,10),(10,5)

$${m}^{\mathrm{6}} +{n}^{\mathrm{6}} +\mathrm{375}{m}^{\mathrm{2}} {n}^{\mathrm{2}} =\mathrm{1953125} \\ $$$$\Rightarrow{m}^{\mathrm{6}} +{n}^{\mathrm{6}} +\mathrm{3}{m}^{\mathrm{2}} {n}^{\mathrm{2}} \left(\mathrm{125}\right)=\left(\mathrm{125}\right)^{\mathrm{3}} \\ $$$${Compare}\:{it}\:{with}: \\ $$$$\:\:\:\left({m}^{\mathrm{2}} \right)^{\mathrm{3}} +\left({n}^{\mathrm{2}} \right)^{\mathrm{3}} +\mathrm{3}{m}^{\mathrm{2}} {n}^{\mathrm{2}} \left({m}^{\mathrm{2}} +{n}^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({m}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)^{\mathrm{3}} \\ $$$$\Rightarrow{m}^{\mathrm{2}} +{n}^{\mathrm{2}} =\mathrm{125} \\ $$$$\:\:\:\:\:{m}=\sqrt{\mathrm{125}−{n}^{\mathrm{2}} } \\ $$$${Only}\:{for}\:{n}=\mathrm{2},\mathrm{11},\mathrm{5},\mathrm{10}\:\:\:{m}\in\mathbb{Z}^{+} \\ $$$${Hence} \\ $$$${The}\:{solutions}\:{are} \\ $$$$\left({m},{n}\right)=\left(\mathrm{2},\mathrm{11}\right),\left(\mathrm{11},\mathrm{2}\right),\left(\mathrm{5},\mathrm{10}\right),\left(\mathrm{10},\mathrm{5}\right) \\ $$

Commented by mr W last updated on 21/Sep/19

nice solution!

$${nice}\:{solution}! \\ $$

Commented by Rasheed.Sindhi last updated on 21/Sep/19

Thαnks Sir!

$$\mathcal{T}{h}\alpha{nks}\:{Sir}! \\ $$

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