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Question Number 94601 by I want to learn more last updated on 19/May/20

	Answered by Rasheed.Sindhi last updated on 20/May/20
	$α+β+γ=6...........................(i) α^3 +β^3 +γ^3 =87......................(ii) (α+1)(β+1)(γ+1)=33.........(iii) (1/α)+(1/β)+(1/γ)=(m/n) [ gcd(m,n)=1] m+n=? ∨∧∨∧∨∧∨∧∨∧∨∧∨∧∨∧∨∧∨ (m/n)=(1/α)+(1/β)+(1/γ)=((αβ+βγ+γα)/(αβγ))=? So we need two quantities: αβγ & αβ+βγ+γα ^• Let′s use formula to determine a relation between these quantities: (α+β+γ)(α^2 +β^2 +γ^2 −αβ−βγ−γα) =α^3 +β^3 +γ^3 −3αβγ (α+β+γ){(α+β+γ)^2 −3αβ−3βγ−3γα} =α^3 +β^3 +γ^3 −3αβγ (6)(6^2 −3(αβ+βγ+γα))=87−3αβγ 72−6(αβ+βγ+γα)=29−αβγ αβγ−6(αβ+βγ+γα)=−43 let αβγ=p & αβ+βγ+γα=q p−6q=−43.........................A ^• Another relation can be derived from (iii) (iii)⇒αβγ+(αβ+βγ+γα)+(α+β+γ)+1=33 αβγ+(αβ+βγ+γα)=26 p+q=26............................B ^(• ) From A & B: −7q=−69 q=((69)/7) p=26−q=26−((69)/7)=((182−69)/7)=((113)/7) (q/p)=((113/7)/(69/7))=((113)/(69))=(m/n) ^(Note that 113 & 69 are coprime.) Hence m+n=113+69=182$
	$α + β + γ = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . (i)$ $α^{3} + β^{3} + γ^{3} = 87 . . . . . . . . . . . . . . . . . . . . . . (i i)$ $(α + 1) (β + 1) (γ + 1) = 33 . . . . . . . . . (i i i)$ $\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = \frac{m}{n} [g c d (m, n) = 1]$ $m + n = ?$ $\lor \land \lor \land \lor \land \lor \land \lor \land \lor \land \lor \land \lor \land \lor \land \lor$ $\frac{m}{n} = \frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = \frac{α β + β γ + γ α}{α β γ} = ?$ $S o w e n e e d t w o q u a n t i t i e s :$ $α β γ & α β + β γ + γ α$ $^{∙} {L e t}^{'} s u s e f o r m u l a t o d e t e r m i n e$ $a r e l a t i o n b e t w e e n t h e s e q u a n t i t i e s :$ $(α + β + γ) (α^{2} + β^{2} + γ^{2} - α β - β γ - γ α)$ $= α^{3} + β^{3} + γ^{3} - 3 α β γ$ $(α + β + γ) {{(α + β + γ)}^{2} - 3 α β - 3 β γ - 3 γ α}$ $= α^{3} + β^{3} + γ^{3} - 3 α β γ$ $(6) (6^{2} - 3 (α β + β γ + γ α)) = 87 - 3 α β γ$ $72 - 6 (α β + β γ + γ α) = 29 - α β γ$ $α β γ - 6 (α β + β γ + γ α) = - 43$ $l e t α β γ = p & α β + β γ + γ α = q$ $p - 6 q = - 43 . . . . . . . . . . . . . . . . . . . . . . . . . A$ $^{∙} A n o t h e r r e l a t i o n c a n b e d e r i v e d$ $f r o m (i i i)$ $(i i i) \Rightarrow α β γ + (α β + β γ + γ α) + (α + β + γ) + 1 = 33$ $α β γ + (α β + β γ + γ α) = 26$ $p + q = 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . B$ $^{∙} F r o m A & B :$ $- 7 q = - 69$ $q = \frac{69}{7}$ $p = 26 - q = 26 - \frac{69}{7} = \frac{182 - 69}{7} = \frac{113}{7}$ $\frac{q}{p} = \frac{113 / 7}{69 / 7} = \frac{113}{69} = \frac{m}{n}$ $^{N o t e t h a t 113 & 69 a r e c o p r i m e .}$ $H e n c e$ $m + n = 113 + 69 = 182$
	Commented by I want to learn more last updated on 20/May/20

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