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Question Number 165442 by HongKing last updated on 01/Feb/22

Answered by Rasheed.Sindhi last updated on 02/Feb/22

$$\{\begin{array}{l}25725-a^2=b^3\\ c^2-25725=d^2\end{array};a,b,c,d\in {\mathbb{Z}}^{+}$$$$\underset{-}{\mathcal{T}hesmallesta+b+c+d=?}$$$$\mathcal{T}hesmallesta+b+c+d$$$$=(\mathcal{T}hesmallesta+b)$$$$+(\mathcal{T}hesmallestc+d)$$$$\bullet 25725-a^2=b^3\Rightarrow a=\sqrt{25725-b^3}>0$$$$\Rightarrow 25725-b^3>0\Rightarrow b^3<25725$$$$\Rightarrow b<\sqrt[3]{25725}=29.52\Rightarrow b⩽29$$$$b=5\Rightarrow a=160\left(theonlypositiveintegralsolution\right)$$$$\underset{-}{\therefore \mathcal{T}hesmallesta+b=160+5=165}$$$$\bullet c^2-25725=d^2\Rightarrow c^2-d^2=25725$$$$\Rightarrow (c-d)(c+d)=25725$$$$\Rightarrow c+d=\frac{25725}{c-d}$$$$\mathcal{T}hesmallestc+d\Rightarrow \mathcal{T}hegreatestc-d$$$$\mathcal{T}hegreatestc-d⩽\lfloor \sqrt{25725}\rfloor =160$$$$\mathcal{T}hegreatestc-d=147$$$$\underset{-}{\therefore \mathcal{T}hesmallestc+d=25765/147=175}$$$$$$$$\stackrel{\bullet }{\bullet \bullet }\mathcal{T}hesmallesta+b+c+d$$$$=\left(165\right)+\left(175\right)=340$$

Commented by HongKing last updated on 06/Feb/22

$$\mathrm{cool}\mathrm{dear}\mathrm{Sir}\mathrm{thank}\mathrm{you}$$

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