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Question Number 155480 by mathdanisur last updated on 01/Oct/21
Find the positive integer solution  of the equation:  x^3  + y^3  = 911(xy + 49)
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{911}\left(\mathrm{xy}\:+\:\mathrm{49}\right) \\ $$
Answered by Rasheed.Sindhi last updated on 01/Oct/21
  x^3  + y^3  = 911(xy + 49)  (x+y)(x^2 −xy+y^2 )=911(xy+49)  x+y=911 ∧ x^2 −xy+y^2 =xy+49  x+y=911 ∧ (x−y)^2 =49  x+y=911 ∧ x−y=±7  2x=911±7  x=((911±7)/2)=459,452  x=459:   y=911−x=911−459=452  x=452:   y=911−x=911−452=459  (x,y)=(459,452),(452,459)
$$ \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{911}\left(\mathrm{xy}\:+\:\mathrm{49}\right) \\ $$$$\left(\mathrm{x}+\mathrm{y}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{xy}+\mathrm{y}^{\mathrm{2}} \right)=\mathrm{911}\left(\mathrm{xy}+\mathrm{49}\right) \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{911}\:\wedge\:\mathrm{x}^{\mathrm{2}} −\mathrm{xy}+\mathrm{y}^{\mathrm{2}} =\mathrm{xy}+\mathrm{49} \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{911}\:\wedge\:\left(\mathrm{x}−\mathrm{y}\right)^{\mathrm{2}} =\mathrm{49} \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{911}\:\wedge\:\mathrm{x}−\mathrm{y}=\pm\mathrm{7} \\ $$$$\mathrm{2x}=\mathrm{911}\pm\mathrm{7} \\ $$$$\mathrm{x}=\frac{\mathrm{911}\pm\mathrm{7}}{\mathrm{2}}=\mathrm{459},\mathrm{452} \\ $$$$\mathrm{x}=\mathrm{459}:\:\:\:\mathrm{y}=\mathrm{911}−\mathrm{x}=\mathrm{911}−\mathrm{459}=\mathrm{452} \\ $$$$\mathrm{x}=\mathrm{452}:\:\:\:\mathrm{y}=\mathrm{911}−\mathrm{x}=\mathrm{911}−\mathrm{452}=\mathrm{459} \\ $$$$\left(\mathrm{x},\mathrm{y}\right)=\left(\mathrm{459},\mathrm{452}\right),\left(\mathrm{452},\mathrm{459}\right) \\ $$
Commented by mathdanisur last updated on 01/Oct/21
Very nice Ser thank you
$$\mathrm{Very}\:\mathrm{nice}\:\boldsymbol{\mathrm{S}}\mathrm{er}\:\mathrm{thank}\:\mathrm{you} \\ $$
Commented by otchereabdullai@gmail.com last updated on 08/Oct/21
nice
$$\mathrm{nice} \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/21
Thanks!
$${Thanks}! \\ $$

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