Question Number 112528 by dw last updated on 08/Sep/20 $${If}\:\:\:\:\:\frac{{n}−\mathrm{7}}{{n}−\mathrm{17}}\:\:\:\:\:\:\:{and}\:\:\:\:\frac{{n}−\mathrm{14}}{{n}−\mathrm{24}}\:\:\:{are}\:{integers}\:{numbers} \\ $$$${n}=? \\ $$ Commented by Dwaipayan Shikari last updated on 08/Sep/20 $$\mathrm{1}+\frac{\mathrm{10}}{{n}−\mathrm{17}}\:\:\:\:\:\:\:{n}\left(\mathrm{7},\mathrm{12},\mathrm{18},\mathrm{22},\mathrm{27}\right) \\ $$$$\frac{{n}−\mathrm{10}}{{n}−\mathrm{24}}=\mathrm{1}+\frac{\mathrm{14}}{{n}−\mathrm{24}}\:\:\:\:{n}\left(\mathrm{10},\mathrm{31},\mathrm{38}\right)…
Question Number 178054 by Acem last updated on 12/Oct/22 $${Mr}.\:{w}\:{wants}\:{to}\:{distribute}\:{n}+\mathrm{1}\:{different}\: \\ $$$${prizes}\:\:{to}\:{n}\:{friends}\:{so}\:{that}\:{each}\:{one}\:{gets} \\ $$$$\:{at}\:{least}\:{one}\:{prize},\:{how}\:{many}\:{results} \\ $$$$\:{of}\:{this}\:{process}.\:{I}\:{hope}\:{to}\:{be}\:{that}\:{one}\:{gets}\:\mathrm{2} \\ $$$$ \\ $$ Commented by mr W last…
Question Number 46978 by rahul 19 last updated on 03/Nov/18 Answered by MrW3 last updated on 03/Nov/18 $$\frac{{q}}{{p}}=\mathrm{4}−{p}\Rightarrow{p}=\mathrm{4}−\frac{{q}}{{p}} \\ $$$$\frac{{r}}{{q}}=\mathrm{4}−{q}\Rightarrow{q}=\mathrm{4}−\frac{{r}}{{q}} \\ $$$$\frac{{p}}{{r}}=\mathrm{4}−{r}\Rightarrow{r}=\mathrm{4}−\frac{{p}}{{r}} \\ $$$$\Rightarrow{p}+{q}+{r}=\mathrm{12}−\left(\frac{{q}}{{p}}+\frac{{r}}{{q}}+\frac{{p}}{{r}}\right)\leqslant\mathrm{12}−\mathrm{3}×\:^{\mathrm{3}} \sqrt{\frac{{q}}{{p}}×\frac{{r}}{{q}}×\frac{{p}}{{r}}}=\mathrm{12}−\mathrm{3}=\mathrm{9}…
Question Number 46974 by rahul 19 last updated on 03/Nov/18 $${Number}\:{of}\:{integers}\:{n}\:{for}\:{which}\: \\ $$$$\mathrm{3}{x}^{\mathrm{3}} −\mathrm{25}{x}+{n}=\mathrm{0}\:{has}\:{three}\:{real}\:{roots}\:{is}\:? \\ $$$$ \\ $$ Commented by rahul 19 last updated on…
Question Number 112509 by I want to learn more last updated on 08/Sep/20 Commented by mr W last updated on 13/Sep/20 $${solution}\:{see}\:{Q}\mathrm{113486} \\ $$ Terms…
Question Number 46973 by rahul 19 last updated on 03/Nov/18 Answered by MJS last updated on 03/Nov/18 $${x}^{\mathrm{3}} −\mathrm{27}{x}+{k}=\mathrm{0} \\ $$$${D}=\frac{{k}^{\mathrm{2}} }{\mathrm{4}}−\mathrm{729}<\mathrm{0}\:\Rightarrow\:−\mathrm{54}\leqslant{k}\leqslant\mathrm{54} \\ $$$$\mathrm{but}\:\mathrm{we}\:\mathrm{need}\:\mathrm{integer}\:\mathrm{solutions} \\…
Question Number 46959 by rahul 19 last updated on 03/Nov/18 $${The}\:{reminder}\:{when}\:{polynomial} \\ $$$$\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} +{x}^{\mathrm{6}} +….+{x}^{\mathrm{22}} \:{is}\:{divided}\:{by} \\ $$$$\mathrm{1}+{x}^{} +{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +…..+{x}^{\mathrm{11}} \:{is}\:=? \\ $$…
Question Number 178021 by Acem last updated on 12/Oct/22 $$ \\ $$$${Let}\:{S}=\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right\}\:,\:{we}\:{want}\:{to}\:{make}\:{a}\: \\ $$$$\:{group}\:{H}\:{of}\:{numbers}\:{have}\:{the}\:{following}\: \\ $$$$\:{properties}: \\ $$$$\:\mathrm{1}.\:{Each}\:{number}\:{has}\:{different}\:{digits}\:{and} \\ $$$$\:{taken}\:{from}\:{S} \\ $$$$\:\mathrm{2}.\:{Each}\:{number}\:{is}\:{greater}\:{than}\:\mathrm{20}\:\mathrm{000} \\ $$$$\:\mathrm{3}.\:{None}\:{of}\:{them}\:{is}\:{multiple}\:{of}\:{five} \\…
Question Number 112478 by dw last updated on 08/Sep/20 $${If}\:{P}\left({x}\right)=\:{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c},\:{with}\:{a},\:{b}\:{and}\:{c}\:{real}\:{numbers}. \\ $$$${Roots}\:{of}\:{P}\left({x}\right)\:{z}+\mathrm{3}{i},\:{z}+\mathrm{9}{i}\:{and}\:\mathrm{2}{z}−\mathrm{4},\:{find}\:\mid{a}+{b}+{c}\mid. \\ $$$$\boldsymbol{{N}}{ote}:\:{z}\:{is}\:{complex}\:{number}. \\ $$ Answered by floor(10²Eta[1]) last updated on 08/Sep/20…
Question Number 178008 by Spillover last updated on 12/Oct/22 Answered by Beginner last updated on 12/Oct/22 Commented by Spillover last updated on 12/Oct/22 $$\mathrm{thanks}\:\mathrm{for}\:\mathrm{doing}\:\mathrm{it}. \\…