Category: Number Theory

Question Number 199331 by necx122 last updated on 01/Nov/23 $$Whatistheremainderwhen$$$$1^1+2^2+3^3+\dots \dots +{2023}^{2023}isdividedby7$$ Answered by AST last updated on 01/Nov/23…

Question Number 198684 by cortano12 last updated on 23/Oct/23 $$\cancel{\zeta }$$ Answered by Red1ight last updated on 23/Oct/23 $$24,88$$$$A=24\times 88=2112$$$${P}=\mathrm{24}+\mathrm{88}=\mathrm{112} \…

Question Number 198400 by cortano12 last updated on 19/Oct/23 $${20}^{11}-1=\dots \left(\mathrm{mod}1000\right)$$ Answered by MM42 last updated on 19/Oct/23 $$\mathrm{20}^{\mathrm{11}} −\mathrm{1}\overset{\mathrm{1000}} {\equiv}−\mathrm{1}\overset{\mathrm{1000}} {\equiv}\mathrm{999} \…

Question Number 197461 by Erico last updated on 18/Sep/23 $$\mathrm{Prove}\mathrm{that}:$$$$\bullet {\underset{0}{\int }}^{\mathrm{x}}\frac{\mathrm{lnt}}{{\mathrm{t}}^2-1}\mathrm{dt}={\underset{0}{\int }}^{\frac{\pi }{2}}\arctan \left(\mathrm{xtan}\theta \right)\mathrm{d}\theta$$$$\bullet\:\:\underset{\:\frac{\mathrm{1}}{\mathrm{x}}} {\int}^{\:\mathrm{x}} \frac{\mathrm{lnt}}{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\mathrm{arctant}\:\mathrm{dt}=\frac{\pi}{\mathrm{8}}\underset{\:\mathrm{0}} {\int}^{\:\pi} \mathrm{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{sint}\right)\mathrm{dt} \…

Question Number 196672 by Rasheed.Sindhi last updated on 29/Aug/23 $$Findall\mathrm{\Omega }\overline{=abcdef},suchthat$$$$abcdef=abc+def$$ Commented by Rasheed.Sindhi last updated on 29/Aug/23 $$\text{You can't use 'macro parameter character \#' in math mode}$$…