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Question-141453

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Question Number 141453 by Khalmohmmad last updated on 19/May/21
Commented by PRITHWISH SEN 2 last updated on 19/May/21
let ab=t a+b=u then tu=880 t+u=71 ⇒t−u=(√(71^2 −4×880)) = ±39 ⇒t=55,u=16 (cancelling the −ve case a=11,b=5 or a=5,b=11 in either case a^6 +b^6 = 1787186
$$\mathrm{let} \\ $$$$\mathrm{ab}=\mathrm{t}\:\:\:\mathrm{a}+\mathrm{b}=\mathrm{u} \\ $$$$\mathrm{then} \\ $$$$\mathrm{tu}=\mathrm{880} \\ $$$$\mathrm{t}+\mathrm{u}=\mathrm{71} \\ $$$$\Rightarrow\mathrm{t}−\mathrm{u}=\sqrt{\mathrm{71}^{\mathrm{2}} −\mathrm{4}×\mathrm{880}}\:=\:\pm\mathrm{39} \\ $$$$\Rightarrow\mathrm{t}=\mathrm{55},\mathrm{u}=\mathrm{16}\:\:\:\left(\mathrm{cancelling}\:\mathrm{the}\:−\mathrm{ve}\:\mathrm{case}\right. \\ $$$$\mathrm{a}=\mathrm{11},\mathrm{b}=\mathrm{5}\:\:\:\mathrm{or}\:\mathrm{a}=\mathrm{5},\mathrm{b}=\mathrm{11} \\ $$$$\mathrm{in}\:\mathrm{either}\:\mathrm{case} \\ $$$$\mathrm{a}^{\mathrm{6}} +\mathrm{b}^{\mathrm{6}} \:=\:\mathrm{1787186} \\ $$
Answered by Rasheed.Sindhi last updated on 19/May/21
ab(a+b)=880 ∧ ab+a+b=71 ab=71−(a+b) {71−(a+b)}(a+b)=880 (a+b)^2 −71(a+b)+880=0 a+b=((71±(√(71^2 −4(880))))/2) a+b=((71±39)/2)=55,16 ab=71−55,71−16=16,55 (a+b,ab)=(55,16) , (16,55) ^• (a+b,ab)=(55,16) a+b=55 (a+b)^2 =3025 a^2 +b^2 =3025−2ab=3025−2(16)=2993 (a^2 +b^2 )^3 =(2993)^3 a^6 +b^6 +3(ab)^2 (a^2 +b^2 )=2993^3 a^6 +b^6 =2993^3 −3(256)(2993) =26809142033 ^• (a+b,ab)=(16,55) (a+b)^3 =16^3 a^3 +b^3 +3ab(a+b)=16^3 a^3 +b^3 =16^3 −3(55)(16)=1456 (a^3 +b^3 )^2 =1456^2 a^6 +b^6 =1456^2 −2(55)^3 =1787186
$${ab}\left({a}+{b}\right)=\mathrm{880}\:\wedge\:{ab}+{a}+{b}=\mathrm{71} \\ $$$${ab}=\mathrm{71}−\left({a}+{b}\right) \\ $$$$\left\{\mathrm{71}−\left({a}+{b}\right)\right\}\left({a}+{b}\right)=\mathrm{880} \\ $$$$\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{71}\left({a}+{b}\right)+\mathrm{880}=\mathrm{0} \\ $$$${a}+{b}=\frac{\mathrm{71}\pm\sqrt{\mathrm{71}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{880}\right)}}{\mathrm{2}} \\ $$$${a}+{b}=\frac{\mathrm{71}\pm\mathrm{39}}{\mathrm{2}}=\mathrm{55},\mathrm{16} \\ $$$${ab}=\mathrm{71}−\mathrm{55},\mathrm{71}−\mathrm{16}=\mathrm{16},\mathrm{55} \\ $$$$\left({a}+{b},{ab}\right)=\left(\mathrm{55},\mathrm{16}\right)\:,\:\left(\mathrm{16},\mathrm{55}\right) \\ $$$$\:^{\bullet} \left({a}+{b},{ab}\right)=\left(\mathrm{55},\mathrm{16}\right) \\ $$$$\:\:\:\:{a}+{b}=\mathrm{55} \\ $$$$\:\:\:\:\left({a}+{b}\right)^{\mathrm{2}} =\mathrm{3025} \\ $$$$\:\:\:\:\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{3025}−\mathrm{2}{ab}=\mathrm{3025}−\mathrm{2}\left(\mathrm{16}\right)=\mathrm{2993} \\ $$$$\:\:\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{3}} =\left(\mathrm{2993}\right)^{\mathrm{3}} \\ $$$$\:\:\:{a}^{\mathrm{6}} +{b}^{\mathrm{6}} +\mathrm{3}\left({ab}\right)^{\mathrm{2}} \left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)=\mathrm{2993}^{\mathrm{3}} \\ $$$$\:\:\:{a}^{\mathrm{6}} +{b}^{\mathrm{6}} =\mathrm{2993}^{\mathrm{3}} −\mathrm{3}\left(\mathrm{256}\right)\left(\mathrm{2993}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{26809142033} \\ $$$$\:^{\bullet} \left({a}+{b},{ab}\right)=\left(\mathrm{16},\mathrm{55}\right) \\ $$$$\:\:\left({a}+{b}\right)^{\mathrm{3}} =\mathrm{16}^{\mathrm{3}} \\ $$$$\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +\mathrm{3}{ab}\left({a}+{b}\right)=\mathrm{16}^{\mathrm{3}} \\ $$$$\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\mathrm{16}^{\mathrm{3}} −\mathrm{3}\left(\mathrm{55}\right)\left(\mathrm{16}\right)=\mathrm{1456} \\ $$$$\:\:\:\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} \right)^{\mathrm{2}} =\mathrm{1456}^{\mathrm{2}} \\ $$$$\:\:\:{a}^{\mathrm{6}} +{b}^{\mathrm{6}} =\mathrm{1456}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{55}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1787186} \\ $$
Answered by Rasheed.Sindhi last updated on 19/May/21
a^6 +b^6 =(a^2 +b^2 )(a^4 −a^2 b^2 +b^4 ) =(a^2 +b^2 )( (a^2 +b^2 )^2 −3a^2 b^2 ) =((a+b)^2 −2ab)( ((a+b)^2 −2ab)^2 −3(ab)^2 ) ■a+b=55,ab=16 (imported) =( 55^2 −2(16) )( (55^2 −2(16))^2 −3(16)^2 ) =(2993)( 2993^2 −3(256) ) =26809142033 ■ a+b=16 , ab=55 (imported) a^6 +b^6 =((a+b)^2 −2ab)( ((a+b)^2 −2ab)^2 −3(ab)^2 ) =(16^2 −2(55))( 16^2 −2(55) )^2 −3(55)^2 ) =146(146^2 −3×55^2 ) =1787186
$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} \\ $$$$=\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\left({a}^{\mathrm{4}} −{a}^{\mathrm{2}} {b}^{\mathrm{2}} +{b}^{\mathrm{4}} \right) \\ $$$$=\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\left(\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{2}} −\mathrm{3}{a}^{\mathrm{2}} {b}^{\mathrm{2}} \right) \\ $$$$=\left(\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{2}{ab}\right)\left(\:\left(\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{2}{ab}\right)^{\mathrm{2}} −\mathrm{3}\left({ab}\right)^{\mathrm{2}} \right) \\ $$$$\:\:\blacksquare{a}+{b}=\mathrm{55},{ab}=\mathrm{16}\:\left({imported}\right) \\ $$$$=\left(\:\mathrm{55}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{16}\right)\:\right)\left(\:\left(\mathrm{55}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{16}\right)\right)^{\mathrm{2}} −\mathrm{3}\left(\mathrm{16}\right)^{\mathrm{2}} \right) \\ $$$$=\left(\mathrm{2993}\right)\left(\:\mathrm{2993}^{\mathrm{2}} −\mathrm{3}\left(\mathrm{256}\right)\:\right) \\ $$$$=\mathrm{26809142033} \\ $$$$\:\:\blacksquare\:{a}+{b}=\mathrm{16}\:,\:{ab}=\mathrm{55}\:\left({imported}\right) \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} \\ $$$$=\left(\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{2}{ab}\right)\left(\:\left(\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{2}{ab}\right)^{\mathrm{2}} −\mathrm{3}\left({ab}\right)^{\mathrm{2}} \right) \\ $$$$\left.=\left(\mathrm{16}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{55}\right)\right)\left(\:\mathrm{16}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{55}\right)\:\right)^{\mathrm{2}} −\mathrm{3}\left(\mathrm{55}\right)^{\mathrm{2}} \right) \\ $$$$=\mathrm{146}\left(\mathrm{146}^{\mathrm{2}} −\mathrm{3}×\mathrm{55}^{\mathrm{2}} \right) \\ $$$$=\mathrm{1787186} \\ $$
Answered by MJS_new last updated on 19/May/21
a=x−y∧b=x+y 2x^3 −2xy^2 =880 ⇒ y^2 =((x^3 −440)/x) x^2 +2x−y^2 =71 ⇒ y^2 =x^2 +2x−71 ((x^3 −440)/x)=x^2 +2x−71 x^2 −((71)/2)x+220=0 ⇒ x=8∨x=((55)/2) ⇒ y^2 =9∨y^2 =((2961)/4) a^6 +b^6 =2x^6 +30x^4 y^2 +30x^2 y^4 +2y^6 a^6 +b^6 =1787186∨26809142033
$${a}={x}−{y}\wedge{b}={x}+{y} \\ $$$$\mathrm{2}{x}^{\mathrm{3}} −\mathrm{2}{xy}^{\mathrm{2}} =\mathrm{880}\:\Rightarrow\:{y}^{\mathrm{2}} =\frac{{x}^{\mathrm{3}} −\mathrm{440}}{{x}} \\ $$$${x}^{\mathrm{2}} +\mathrm{2}{x}−{y}^{\mathrm{2}} =\mathrm{71}\:\Rightarrow\:{y}^{\mathrm{2}} ={x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{71} \\ $$$$\frac{{x}^{\mathrm{3}} −\mathrm{440}}{{x}}={x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{71} \\ $$$${x}^{\mathrm{2}} −\frac{\mathrm{71}}{\mathrm{2}}{x}+\mathrm{220}=\mathrm{0}\:\Rightarrow\:{x}=\mathrm{8}\vee{x}=\frac{\mathrm{55}}{\mathrm{2}} \\ $$$$\Rightarrow\:{y}^{\mathrm{2}} =\mathrm{9}\vee{y}^{\mathrm{2}} =\frac{\mathrm{2961}}{\mathrm{4}} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} =\mathrm{2}{x}^{\mathrm{6}} +\mathrm{30}{x}^{\mathrm{4}} {y}^{\mathrm{2}} +\mathrm{30}{x}^{\mathrm{2}} {y}^{\mathrm{4}} +\mathrm{2}{y}^{\mathrm{6}} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} =\mathrm{1787186}\vee\mathrm{26809142033} \\ $$

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