Question Number 203146 by Rasheed.Sindhi last updated on 11/Jan/24 $$\mathcal{D}{etermine}\:\overline {{ab}}\:\left({a}>{b}\right)\:{such}\:{that}\left(\:\overline {{ab}}+\overline {{ba}}\right)\: \\ $$$${and}\:\left(\overline {{ab}}−\overline {{ba}}\right)\:{are}\:{both}\:{perfect}\:{squares}. \\ $$ Answered by Frix last updated on…
Question Number 202716 by Rasheed.Sindhi last updated on 01/Jan/24 $$\overline {\:\:{abcd}\:\:}{is}\:{a}\:{four}\:{digit}\:{number} \\ $$$${such}\:{that}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} =\overline {\:{cd}\:} \\ $$$${and}\:\overline {\:{cd}\:}−\overline {\:{d}\:}=\overline {\:{ab}\:}. \\ $$$$\mathcal{F}{ind}\:{the}\:{number}.…
Question Number 201418 by cortano12 last updated on 06/Dec/23 $$\:\:\:\:\:\:\mathrm{2025}^{\mathrm{2025}} \:=\:\mathrm{x}\:\left(\mathrm{mod}\:\mathrm{17}\:\right) \\ $$ Answered by mr W last updated on 06/Dec/23 $$\mathrm{2025}^{\mathrm{2025}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$=\left(\mathrm{119}×\mathrm{17}+\mathrm{2}\right)^{\mathrm{2025}}…
Question Number 201352 by cortano12 last updated on 05/Dec/23 $$\:\:\:\mathrm{2023}^{\mathrm{2023}} \:=\:…\:\left(\mathrm{mod}\:\mathrm{13}\right) \\ $$ Answered by Rasheed.Sindhi last updated on 05/Dec/23 $$\:\:\:\mathrm{2023}^{\mathrm{2023}} \:\equiv\:…\:\left(\mathrm{mod}\:\mathrm{13}\right) \\ $$$$\mathrm{2023}^{\mathrm{2023}} \\…
Question Number 200836 by Rasheed.Sindhi last updated on 24/Nov/23 $$ \\ $$$$\mathcal{L}{et}\overline {\:{abc}\:}+\overline {\:{bca}\:}+\overline {\:{cab}\:}=\overline {\:{defg}\:} \\ $$$${where}\:{a},{b},…,{g}\:{are}\:{decimal}\:{digits} \\ $$$$\left({may}\:{be}\:{equal}\:{to}\:\mathrm{0}\right)\: \\ $$$${Show}\:{that} \\ $$$$\left({i}\right)\overline {\:{dg}\:}={a}+{b}+{c}…
Question Number 200834 by Rasheed.Sindhi last updated on 24/Nov/23 $${Show}\:{that}\:\:\:\overline {\:\:{abc}\:}+\overline {\:{bca}\:}+\overline {\:{cab}\:} \\ $$$${is}\:{divisible}\:{by}\:\mathrm{37} \\ $$ Answered by AST last updated on 24/Nov/23 $$\mathrm{111}\left({a}+{b}+{c}\right)\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{37}…
Question Number 200315 by Rasheed.Sindhi last updated on 17/Nov/23 $$\:\begin{cases}{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}+\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{cde}\:}}\\{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}−\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{f}\:}\:}\end{cases} \\ $$$${a},{b},{c},{d},{e},{f}\:{are}\:{all}\:{different}\:{and}\:{in} \\ $$$${some}\:{order}\:{consecutive}\:{also}. \\…
Question Number 200200 by Fridunatjan08 last updated on 16/Nov/23 $${Find}\:{four}\:{positive}\:{integers}, \\ $$$$\:{each}\:{not}\:{exceeding}\:\mathrm{70000}\:{and}\: \\ $$$${each}\:{having}\:{more}\:{than}\:\mathrm{100} \\ $$$$\:{divisors}. \\ $$ Answered by mr W last updated on…
Question Number 200041 by depressiveshrek last updated on 12/Nov/23 $${By}\:{strong}\:{induction}\:{prove}\:{that}\:{any} \\ $$$${natural}\:{number}\:{equal}\:{to}\:{or}\:{bigger}\:{than} \\ $$$$\mathrm{8}\:{can}\:{be}\:{written}\:{as}\:\mathrm{3}{a}+\mathrm{5}{b}\:{where}\:{a}\:{and}\:{b} \\ $$$${are}\:{non}−{negative}\:{integers}. \\ $$ Answered by des_ last updated on 12/Nov/23…
Question Number 199864 by cortano12 last updated on 10/Nov/23 $$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{2019}} {\sum}}\mathrm{n}^{\mathrm{4}} \:\mathrm{when} \\ $$$$\:\:\mathrm{divide}\:\mathrm{by}\:\mathrm{53}\: \\ $$ Answered by AST last updated on 10/Nov/23 $$\underset{{n}=\mathrm{1}}…