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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

{ ((25725−a^2 =b^3 )),((c^2 −25725=d^2 )) :} ; a,b,c,d∈Z^+ The smallest a+b+c+d =? _(−) The smallest a+b+c+d =(The smallest a+b) +(The smallest c+d) • 25725−a^2 =b^3 ⇒a=(√(25725−b^3 ))>0 ⇒25725−b^3 >0⇒b^3 <25725 ⇒b<((25725))^(1/3) =29.52⇒b≤29 b=5⇒a=160 (the only positive integral solution) ∴ The smallest a+b=160+5=165 _(−) •c^2 −25725=d^2 ⇒c^2 −d^2 =25725 ⇒(c−d)(c+d)=25725 ⇒c+d=((25725)/(c−d)) The smallest c+d⇒The greatest c−d The greatest c−d≤⌊(√(25725)) ⌋=160 The greatest c−d=147 ∴The smallest c+d=25765/147=175_(−) • •^(•) The smallest a+b+c+d =(165)+(175)=340

$$\begin{cases}{\mathrm{25725}−{a}^{\mathrm{2}} ={b}^{\mathrm{3}} }\\{{c}^{\mathrm{2}} −\mathrm{25725}={d}^{\mathrm{2}} }\end{cases}\:\:;\:\:{a},{b},{c},{d}\in\mathbb{Z}^{+} \\ $$$$\underset{−} {\mathcal{T}{he}\:{smallest}\:{a}+{b}+{c}+{d}\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \\ $$$$\mathcal{T}{he}\:{smallest}\:{a}+{b}+{c}+{d} \\ $$$$\:\:\:\:\:\:\:=\left(\mathcal{T}{he}\:{smallest}\:{a}+{b}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left(\mathcal{T}{he}\:{smallest}\:{c}+{d}\right) \\ $$$$\bullet\:\mathrm{25725}−{a}^{\mathrm{2}} ={b}^{\mathrm{3}} \Rightarrow{a}=\sqrt{\mathrm{25725}−{b}^{\mathrm{3}} }>\mathrm{0} \\ $$$$\Rightarrow\mathrm{25725}−{b}^{\mathrm{3}} >\mathrm{0}\Rightarrow{b}^{\mathrm{3}} <\mathrm{25725} \\ $$$$\Rightarrow{b}<\sqrt[{\mathrm{3}}]{\mathrm{25725}}\:=\mathrm{29}.\mathrm{52}\Rightarrow{b}\leqslant\mathrm{29} \\ $$$${b}=\mathrm{5}\Rightarrow{a}=\mathrm{160}\:\left({the}\:{only}\:{positive}\:{integral}\:{solution}\right) \\ $$$$\underset{−} {\therefore\:\mathcal{T}{he}\:{smallest}\:{a}+{b}=\mathrm{160}+\mathrm{5}=\mathrm{165}\:\:\:\:} \\ $$$$\bullet{c}^{\mathrm{2}} −\mathrm{25725}={d}^{\mathrm{2}} \Rightarrow{c}^{\mathrm{2}} −{d}^{\mathrm{2}} =\mathrm{25725} \\ $$$$\Rightarrow\left({c}−{d}\right)\left({c}+{d}\right)=\mathrm{25725} \\ $$$$\:\Rightarrow{c}+{d}=\frac{\mathrm{25725}}{{c}−{d}} \\ $$$$\mathcal{T}{he}\:{smallest}\:{c}+{d}\Rightarrow\mathcal{T}{he}\:{greatest}\:{c}−{d} \\ $$$$\mathcal{T}{he}\:{greatest}\:{c}−{d}\leqslant\lfloor\sqrt{\mathrm{25725}}\:\rfloor=\mathrm{160} \\ $$$$\mathcal{T}{he}\:{greatest}\:{c}−{d}=\mathrm{147}\:\:\:\: \\ $$$$\underset{−} {\therefore\mathcal{T}{he}\:{smallest}\:{c}+{d}=\mathrm{25765}/\mathrm{147}=\mathrm{175}} \\ $$$$ \\ $$$$\overset{\bullet} {\bullet\:\:\bullet}\:\:\mathcal{T}{he}\:{smallest}\:{a}+{b}+{c}+{d} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{165}\right)+\left(\mathrm{175}\right)=\mathrm{340} \\ $$

Commented by HongKing last updated on 06/Feb/22

cool dear Sir thank you

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

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