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 $$\underset{{n}=\mathrm{1}}…
Question Number 199311 by necx122 last updated on 01/Nov/23 $${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${that}\:{are}\:{factors}\:{of}\:\mathrm{3}^{\mathrm{19}} .\mathrm{7}^{\mathrm{12}} .\mathrm{10}^{\mathrm{25}} \:{and}\:{are} \\ $$$${also}\:{multiples}\:{of}\:\mathrm{3}^{\mathrm{15}} .\mathrm{7}^{\mathrm{10}} .\mathrm{10}^{\mathrm{19}} \\ $$ Answered by AST last…
Question Number 199331 by necx122 last updated on 01/Nov/23 $${What}\:{is}\:{the}\:{remainder}\:{when} \\ $$$$\mathrm{1}^{\mathrm{1}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{3}} +……+\mathrm{2023}^{\mathrm{2023}} \:{is}\:{divided}\:{by}\:\mathrm{7} \\ $$ Answered by AST last updated on 01/Nov/23…
Question Number 199011 by necx122 last updated on 26/Oct/23 $${Sum}\:{of}\:{two}\:{irrational}\:{numbers}\:{is}\:\mathrm{1} \\ $$$${less}\:{than}\:{their}\:{product},\:{and}\:\mathrm{8}\:{less}\:{than} \\ $$$${their}\:{sum}\:{of}\:{squares}.\:{Find}\:{the}\:{larger} \\ $$$${of}\:{the}\:{two}\:{numbers}. \\ $$ Commented by nikif99 last updated on 27/Oct/23…