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Question Number 156604 by amin96 last updated on 13/Oct/21

Commented by amin96 last updated on 13/Oct/21

help mi pleas

$${help}\:{mi}\:{pleas} \\ $$

Commented by Rasheed.Sindhi last updated on 13/Oct/21

One solution: {a,b,c}={1,2,3}

$$\mathrm{One}\:\:\mathrm{solution}:\:\left\{\mathrm{a},\mathrm{b},\mathrm{c}\right\}=\left\{\mathrm{1},\mathrm{2},\mathrm{3}\right\} \\ $$

Answered by Rasheed.Sindhi last updated on 13/Oct/21

a^3 +b^3 +c^3 =a^2 b^2 c^2 { ((a^2 ∣ b^3 +c^3 ⇒b^3 +c^3 =la^2 )),((b^2 ∣ a^3 +c^3 ⇒a^3 +c^3 =mb^2 )),((c^2 ∣ a^3 +b^3 ⇒a^3 +b^3 =nc^2 )) :} (a^3 +b^3 )+(b^3 +c^3 )+(c^3 +a^3 )=2a^2 b^2 c^2 la^2 +mb^2 +nc^2 =2a^2 b^2 c^2 ..... ... (1/(ab))+(1/(bc))+(1/(ca))=abc ..... ...

$$\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} =\mathrm{a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} \\ $$$$\begin{cases}{\mathrm{a}^{\mathrm{2}} \mid\:\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \Rightarrow\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} ={l}\mathrm{a}^{\mathrm{2}} }\\{\mathrm{b}^{\mathrm{2}} \mid\:\mathrm{a}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \Rightarrow\mathrm{a}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} ={m}\mathrm{b}^{\mathrm{2}} }\\{\mathrm{c}^{\mathrm{2}} \mid\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} \Rightarrow\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} ={n}\mathrm{c}^{\mathrm{2}} }\end{cases} \\ $$$$\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} \right)+\left(\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \right)+\left(\mathrm{c}^{\mathrm{3}} +\mathrm{a}^{\mathrm{3}} \right)=\mathrm{2a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} \\ $$$${l}\mathrm{a}^{\mathrm{2}} +{m}\mathrm{b}^{\mathrm{2}} +{n}\mathrm{c}^{\mathrm{2}} =\mathrm{2a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} \\ $$$$..... \\ $$$$... \\ $$$$\frac{\mathrm{1}}{\mathrm{ab}}+\frac{\mathrm{1}}{\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{ca}}=\mathrm{abc} \\ $$$$..... \\ $$$$... \\ $$$$ \\ $$

Commented by amin96 last updated on 13/Oct/21

thanks sir

$${thanks}\:{sir}\: \\ $$

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