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Question Number 1557 by 123456 last updated on 19/Aug/15

find α,β,γ such that  α=β^2   β^3 =γ^4   γ^5 =α^6

$$\mathrm{find}\:\alpha,\beta,\gamma\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha=\beta^{\mathrm{2}} \\ $$$$\beta^{\mathrm{3}} =\gamma^{\mathrm{4}} \\ $$$$\gamma^{\mathrm{5}} =\alpha^{\mathrm{6}} \\ $$

Commented by 123456 last updated on 19/Aug/15

 { ((α=β^2 )),((β^3 =γ^4 )),((γ^5 =α^6 )) :}⇒ { ((α^(15) =β^(30) )),((β^(30) =γ^(40) )),((γ^(40) =α^(48) )) :}  [(1,2),(3,4),(5,6)]  [(3,6),(6,8),(5,6)]  [(15,30),(30,40),(40,48)]  α^(15) =α^(48) ⇒α^(15) (α^(33) −1)=0

$$\begin{cases}{\alpha=\beta^{\mathrm{2}} }\\{\beta^{\mathrm{3}} =\gamma^{\mathrm{4}} }\\{\gamma^{\mathrm{5}} =\alpha^{\mathrm{6}} }\end{cases}\Rightarrow\begin{cases}{\alpha^{\mathrm{15}} =\beta^{\mathrm{30}} }\\{\beta^{\mathrm{30}} =\gamma^{\mathrm{40}} }\\{\gamma^{\mathrm{40}} =\alpha^{\mathrm{48}} }\end{cases} \\ $$$$\left[\left(\mathrm{1},\mathrm{2}\right),\left(\mathrm{3},\mathrm{4}\right),\left(\mathrm{5},\mathrm{6}\right)\right] \\ $$$$\left[\left(\mathrm{3},\mathrm{6}\right),\left(\mathrm{6},\mathrm{8}\right),\left(\mathrm{5},\mathrm{6}\right)\right] \\ $$$$\left[\left(\mathrm{15},\mathrm{30}\right),\left(\mathrm{30},\mathrm{40}\right),\left(\mathrm{40},\mathrm{48}\right)\right] \\ $$$$\alpha^{\mathrm{15}} =\alpha^{\mathrm{48}} \Rightarrow\alpha^{\mathrm{15}} \left(\alpha^{\mathrm{33}} −\mathrm{1}\right)=\mathrm{0} \\ $$

Answered by Rasheed Soomro last updated on 20/Aug/15

α=β^2  ⇒α^3 =β^6 ..............(I)  β^3 =γ^4  ⇒β^6 =γ^8  ............(II)  From (I)   and   (II)     α^3 =γ^(8  ) ⇒α^6 =γ^(16)  .....(III)  γ^5 =α^6  ⇒α^6 =γ^5 .........................(IV)  From (III)  and    (IV)  γ^5 =γ^(16) ⇒γ^(16) −γ^5 =0⇒ γ^5 (γ^(11) −1)=0  γ=0 ∨ γ^(11) =1 , γ  is 11th  root of unity.  [Above was the process of eliminating   α  and   β.  In the same way  by eliminating   β    and    γ   and  after that  by    α    and    γ     we will get results about   α     and     β  respectively.]  Results about α and  β  α=0 ∨ α^(33) =1 , α  is 33rd  root of unity.  β=0 ∨ β^(33) =1 , β  is 33rd  root of unity.  (α,β,γ)=(0,0,0)  Solution of α or β:  {0}∪{1,w,ω^2 ,...ω^(32) },ω=cos((2π)/(33))+ı sin((2π)/(33))  Solution of  γ: {0}∪{1,ρ,ρ^2 ,...ρ^(10) },ρ=cos((2π)/(11))+ı sin((2π)/(11))  Let (α,β,γ)=(ω^p ,ω^q ,ρ^r ) fulfilling given conditions.  w^p =(ω^q )^2 ⇒ p=2q................................I  (ω^q )^3 =(ρ^r )^4 ⇒ω^(3q) =ρ^(4r) ⇒ω^(15q) =ρ^(20r) ...........II  (ρ^r )^5 =(ω^p )^6 ⇒ρ^(5r) =ω^(6p) ⇒ρ^(20r) =ω^(24p) ..........III  p=2q.......[from I]  24p=15q⇒8p=5q...............[from II  and  III]  8(2q)=5q⇒q=0⇒p=2(0)=0⇒p=0  ρ^(20r) =ω^(24p) =ω^(24(0)) =1=ρ^0  ⇒ 20r=0 ⇒r=0  Sol. Set : {(0,0,0),(1,1,1)}

$$\alpha=\beta^{\mathrm{2}} \:\Rightarrow\alpha^{\mathrm{3}} =\beta^{\mathrm{6}} ..............\left(\boldsymbol{{I}}\right) \\ $$$$\beta^{\mathrm{3}} =\gamma^{\mathrm{4}} \:\Rightarrow\beta^{\mathrm{6}} =\gamma^{\mathrm{8}} \:............\left(\boldsymbol{{II}}\right) \\ $$$${From}\:\left(\boldsymbol{{I}}\right)\:\:\:{and}\:\:\:\left(\boldsymbol{{II}}\right)\:\:\:\:\:\alpha^{\mathrm{3}} =\gamma^{\mathrm{8}\:\:} \Rightarrow\alpha^{\mathrm{6}} =\gamma^{\mathrm{16}} \:.....\left(\boldsymbol{{III}}\right) \\ $$$$\gamma^{\mathrm{5}} =\alpha^{\mathrm{6}} \:\Rightarrow\alpha^{\mathrm{6}} =\gamma^{\mathrm{5}} .........................\left(\boldsymbol{{IV}}\right) \\ $$$${From}\:\left(\boldsymbol{{III}}\right)\:\:{and}\:\:\:\:\left(\boldsymbol{{IV}}\right) \\ $$$$\gamma^{\mathrm{5}} =\gamma^{\mathrm{16}} \Rightarrow\gamma^{\mathrm{16}} −\gamma^{\mathrm{5}} =\mathrm{0}\Rightarrow\:\gamma^{\mathrm{5}} \left(\gamma^{\mathrm{11}} −\mathrm{1}\right)=\mathrm{0} \\ $$$$\gamma=\mathrm{0}\:\vee\:\gamma^{\mathrm{11}} =\mathrm{1}\:,\:\gamma\:\:{is}\:\mathrm{11}\boldsymbol{{th}}\:\:{root}\:{of}\:{unity}. \\ $$$$\left[{Above}\:{was}\:{the}\:{process}\:{of}\:{eliminating}\:\:\:\alpha\:\:{and}\:\:\:\beta.\right. \\ $$$${In}\:{the}\:{same}\:{way}\:\:{by}\:{eliminating}\:\:\:\beta\:\:\:\:{and}\:\:\:\:\gamma\:\:\:{and}\:\:{after}\:{that} \\ $$$${by}\:\:\:\:\alpha\:\:\:\:{and}\:\:\:\:\gamma\:\:\:\:\:{we}\:{will}\:{get}\:{results}\:{about}\:\:\:\alpha\:\:\:\:\:{and}\:\:\:\:\:\beta \\ $$$$\left.{respectively}.\right] \\ $$$${Results}\:{about}\:\alpha\:{and}\:\:\beta \\ $$$$\alpha=\mathrm{0}\:\vee\:\alpha^{\mathrm{33}} =\mathrm{1}\:,\:\alpha\:\:{is}\:\mathrm{33}\boldsymbol{{rd}}\:\:{root}\:{of}\:{unity}. \\ $$$$\beta=\mathrm{0}\:\vee\:\beta^{\mathrm{33}} =\mathrm{1}\:,\:\beta\:\:{is}\:\mathrm{33}\boldsymbol{{rd}}\:\:{root}\:{of}\:{unity}. \\ $$$$\left(\alpha,\beta,\gamma\right)=\left(\mathrm{0},\mathrm{0},\mathrm{0}\right) \\ $$$${Solution}\:{of}\:\alpha\:{or}\:\beta:\:\:\left\{\mathrm{0}\right\}\cup\left\{\mathrm{1},{w},\omega^{\mathrm{2}} ,...\omega^{\mathrm{32}} \right\},\omega={cos}\frac{\mathrm{2}\pi}{\mathrm{33}}+\imath\:{sin}\frac{\mathrm{2}\pi}{\mathrm{33}} \\ $$$${Solution}\:{of}\:\:\gamma:\:\left\{\mathrm{0}\right\}\cup\left\{\mathrm{1},\rho,\rho^{\mathrm{2}} ,...\rho^{\mathrm{10}} \right\},\rho={cos}\frac{\mathrm{2}\pi}{\mathrm{11}}+\imath\:{sin}\frac{\mathrm{2}\pi}{\mathrm{11}} \\ $$$${Let}\:\left(\alpha,\beta,\gamma\right)=\left(\omega^{{p}} ,\omega^{{q}} ,\rho^{{r}} \right)\:{fulfilling}\:{given}\:{conditions}. \\ $$$${w}^{{p}} =\left(\omega^{{q}} \right)^{\mathrm{2}} \Rightarrow\:{p}=\mathrm{2}{q}................................\boldsymbol{{I}} \\ $$$$\left(\omega^{{q}} \right)^{\mathrm{3}} =\left(\rho^{{r}} \right)^{\mathrm{4}} \Rightarrow\omega^{\mathrm{3}{q}} =\rho^{\mathrm{4}{r}} \Rightarrow\omega^{\mathrm{15}{q}} =\rho^{\mathrm{20}{r}} ...........\boldsymbol{{II}} \\ $$$$\left(\rho^{{r}} \right)^{\mathrm{5}} =\left(\omega^{{p}} \right)^{\mathrm{6}} \Rightarrow\rho^{\mathrm{5}{r}} =\omega^{\mathrm{6}{p}} \Rightarrow\rho^{\mathrm{20}{r}} =\omega^{\mathrm{24}{p}} ..........\boldsymbol{{III}} \\ $$$${p}=\mathrm{2}{q}.......\left[{from}\:\boldsymbol{{I}}\right] \\ $$$$\mathrm{24}{p}=\mathrm{15}{q}\Rightarrow\mathrm{8}{p}=\mathrm{5}{q}...............\left[{from}\:\boldsymbol{{II}}\:\:{and}\:\:\boldsymbol{{III}}\right] \\ $$$$\mathrm{8}\left(\mathrm{2}{q}\right)=\mathrm{5}{q}\Rightarrow{q}=\mathrm{0}\Rightarrow{p}=\mathrm{2}\left(\mathrm{0}\right)=\mathrm{0}\Rightarrow{p}=\mathrm{0} \\ $$$$\rho^{\mathrm{20}{r}} =\omega^{\mathrm{24}{p}} =\omega^{\mathrm{24}\left(\mathrm{0}\right)} =\mathrm{1}=\rho^{\mathrm{0}} \:\Rightarrow\:\mathrm{20}{r}=\mathrm{0}\:\Rightarrow{r}=\mathrm{0} \\ $$$${Sol}.\:{Set}\::\:\left\{\left(\mathrm{0},\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\mathrm{1},\mathrm{1}\right)\right\} \\ $$

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