Question Number 225601 by gregori last updated on 04/Nov/25 $$\:\:\: \mathrm{2}^{\mathrm{2025}} ×\mathrm{3}^{\mathrm{2025}} \: \\ $$ Commented by mr W last updated on 04/Nov/25 $${the}\:{unit}\:{digit}\:{of}\:\mathrm{6}^{{n}} {is}\:{always}\:\mathrm{6}.…
Question Number 223480 by Bebe last updated on 26/Jul/25 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 222974 by gabthemathguy25 last updated on 12/Jul/25 Answered by MrGaster last updated on 12/Jul/25 $${P}_{\mathrm{100}} =\mathrm{min}\left\{{x}\in\mathbb{N}\mid\underset{{k}=\mathrm{1}} {\overset{{x}} {\sum}}\underset{{i}=\mathrm{2}} {\overset{\lfloor\sqrt{{k}}\rfloor} {\prod}}\left(\mathrm{1}−\delta\left({k}\:\mathrm{mod}\:{i}\right)\right)\geq\mathrm{101}\right\} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{540}}…
Question Number 222635 by Mingma last updated on 02/Jul/25 Answered by Raphael254 last updated on 28/Jul/25 $${n}\:+\:{S}\left({n}\right)\:+\:{S}\left({S}\left({n}\right)\right)\:=\:\mathrm{2007} \\ $$$$ \\ $$$${n}\:=\:\mathrm{1983} \\ $$$$ \\ $$$$\mathrm{1983}\:+\:\left(\mathrm{1}+\mathrm{9}+\mathrm{8}+\mathrm{3}\right)\:+\:{S}\left(\mathrm{1}+\mathrm{9}+\mathrm{8}+\mathrm{3}\right)\:=…
Question Number 222516 by MrGaster last updated on 29/Jun/25 $$\mathrm{Given}\:\mathrm{the}\:\mathrm{integer}\:{k},\mathrm{how}\:\:\mathrm{to} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{incomplete}\:\mathrm{general} \\ $$$$\mathrm{solution}\:\mathrm{for}\:\mathrm{the}\:\mathrm{non}-\mathrm{trivial}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\:\mathrm{the}\:\mathrm{Diophantine}\:\mathrm{equation}: \\ $$$${a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{ka}^{\mathrm{2}} {b}^{\mathrm{2}} ={c}^{\mathrm{4}} +{d}^{\mathrm{4}} +{kc}^{\mathrm{2}} {d}^{\mathrm{2}}…
Question Number 219732 by Spillover last updated on 01/May/25 Answered by Spillover last updated on 01/May/25 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 219657 by Nicholas666 last updated on 30/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{prove}; \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\left(\mathrm{5}{n}−\mathrm{2}\right)\left(\mathrm{5}{n}−\mathrm{3}\right)}{\left(\mathrm{5}{n}−\mathrm{1}\right)\left(\mathrm{5}{n}−\mathrm{4}\right)}\:=\:\varphi \\ $$$$ \\ $$ Answered by MrGaster last updated…
Question Number 218438 by Marzuk last updated on 10/Apr/25 $${An}\:{amazing}\:{thing}\:{i}\:{saw} \\ $$$${S}\:=\:\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:\mathrm{5}\:+\:\mathrm{6}… \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{2}/\mathrm{2}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{4}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\mathrm{6}/\mathrm{2}….\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\:\mathrm{3}…\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}+\:\mathrm{2}\:+\:\mathrm{3}\:…\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}…\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}{S}\:+\:\mathrm{2}\underset{{n}=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{2}{n}\:+\:\mathrm{1}}{\mathrm{2}} \\ $$$${or},{S}\:−\:\mathrm{2}{S}\:=\:\mathrm{1}\:+\:\underset{{n}=\mathrm{1}} {\overset{\infty}…
Question Number 218208 by Marzuk last updated on 01/Apr/25 $${This}\:{question}\:{is}\:{really}\:{important} \\ $$$${Prove}\:{or}\:{disprove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}^{{n}} {m}+\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{\lceil\frac{{n}}{\mathrm{2}}\rceil} }\:+\:\frac{\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{{n}} }\: \\ $$$$\:{the}\:{limit}\:{exists}\:{for}\:{m}\:\in\:{N}\:\backslash{B} \\ $$$${where}\:{B}\:=\:\left\{{n}\:\mid\:{log}_{\mathrm{2}} \left({n}\right)\:\in\:{N}\:\right\}…
Question Number 218076 by malwan last updated on 28/Mar/25 $${How}\:{many}\:{ways}\:{to}\:{arrnge} \\ $$$${the}\:{letters}\:{ABCCCDEFG} \\ $$$$\left(\mathrm{1}\right)\:{in}\:{general}\:. \\ $$$$\left(\mathrm{2}\right)\:{all}\:\mathrm{3}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{3}\right)\:{only}\:\mathrm{2}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{4}\right)\:{no}\:\mathrm{2}\:{or}\:\mathrm{3}\:{Cs}\:{be}\:{together} \\ $$$$\left(\mathrm{5}\right)\:{no}\:{letter}\:{still}\:\:{in}\:{its} \\ $$$${original}\:{place}\:. \\…