Question Number 58700 by naka3546 last updated on 28/Apr/19
$${a},\:{b},\:{c}\:\:\in\:\:\mathbb{R}^{+} \\ $$$${Find}\:\:{triple}\:\:{of}\:\:{positive}\:\:{real}\:\:{numbers}\:\left({a},\:{b},\:{c}\right)\:\:{that}\:\:{satisfy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{a}\lfloor{b}\rfloor\:\:=\:\:\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{b}\lfloor{c}\rfloor\:\:=\:\:\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{c}\lfloor{a}\rfloor\:\:=\:\:\mathrm{12} \\ $$
Commented by Rasheed.Sindhi last updated on 28/Apr/19
$${In}\:\:{a}\lfloor{b}\rfloor=\mathrm{5}\:,\:\:\lfloor{b}\rfloor\:{is}\:{an}\:{integer} \\ $$$${Hence}\:\:\:{a}\:{is}\:{also}\:{an}\:{integer} \\ $$$${real}×{integer}={integer}\Rightarrow{real}={integer}. \\ $$$${So}\:{a},{b},{c}\:\in\:\mathbb{Z} \\ $$$$\:\:\:\lfloor{a}\rfloor={a}\:,\lfloor{b}\rfloor={b}\:,\:\lfloor{c}\rfloor={c} \\ $$$$\:\:\:\:\:\:{ab}=\mathrm{5} \\ $$$$\:\:\:\:\:\:{bc}=\mathrm{5} \\ $$$$\:\:\:\:\:{ca}=\mathrm{12} \\ $$$$\:\:\:{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} =\mathrm{5}×\mathrm{5}×\mathrm{12}=\mathrm{300}=\mathrm{10}\sqrt{\mathrm{3}}\:???? \\ $$$${But}, \\ $$$$\:{a},{b},{c}\in\mathbb{Z}\Rightarrow{a}^{\mathrm{2}} ,{b}^{\mathrm{2}} ,{c}^{\mathrm{2}} \:\in\mathbb{Z}\Rightarrow{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} \:\in\:\mathbb{Z} \\ $$$${This}\:{contradiction}\:{leads}\:{that}\:{there} \\ $$$${aren}'{t}\:{any}\:{such}\:{a},{b},{c}. \\ $$
Commented by mr W last updated on 28/Apr/19
$${dear}\:{sir},\:{a}×\lfloor{b}\rfloor={integer}\:{doesn}'{t}\:{mean} \\ $$$${that}\:{a}\:{is}\:{integer}.\:{it}\:{means}\:{only}\:{that} \\ $$$${a}\:{is}\:{rational}.\:{e}.{g}.\:{a}=\frac{\mathrm{5}}{\mathrm{3}},\:{b}=\frac{\mathrm{7}}{\mathrm{2}},\:\lfloor{b}\rfloor=\mathrm{3}, \\ $$$$\Rightarrow{a}×\lfloor{b}\rfloor=\mathrm{5}. \\ $$
Commented by Rasheed.Sindhi last updated on 28/Apr/19
$${Sorry}\:{for}\:{deffective}\:{logic}! \\ $$$$\:{You}'{re}\:{very}\:{right}\:{sir}!\:{Thank}\:{you} \\ $$$${to}\:{correct}\:{me}.{Actually}\:{I}'{m}\:{here}\:{to}\:{learn} \\ $$$${from}\:{you}! \\ $$
Commented by mr W last updated on 28/Apr/19
$${You}'{re}\:{welcome}\:{sir}.\:{In}\:{fact}\:{I}\:{have} \\ $$$${learnt}\:{and}\:{still}\:{learn}\:{alot}\:{from}\:{you}\:{sir}. \\ $$$${Many}\:{posts}\:{of}\:{you}\:{have}\:{become}\:{classic} \\ $$$${of}\:{the}\:{forum}.\:{It}'{s}\:{a}\:{pleasure}\:{to}\:{study} \\ $$$${them}. \\ $$
Commented by Rasheed.Sindhi last updated on 29/Apr/19
$$\mathcal{TH}\alpha{nks}\:{to}\:{enhearten}\:{me}\:{in}\:{such}\:{an} \\ $$$${exciting}\:{way}\:{sir}!!! \\ $$