Category: Number Theory

Question Number 218208 by Marzuk last updated on 01/Apr/25 $$Thisquestionisreallyimportant$$$$Proveordisprovethat$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{3^{n}m+3^{n-1}}{2^{\lceil \frac{n}{2}\rceil }}+\frac{3^{n-1}}{2^n}$$$$thelimitexistsform\in N\mathrm{\setminus }B$$$${where}\:{B}\:=\:\left\{{n}\:\mid\:{log}_{\mathrm{2}} \left({n}\right)\:\in\:{N}\:\right\}…

Question Number 217958 by dscm last updated on 23/Mar/25 $$a,b,c\in {\mathbb{Z}}^{+}and$$$$a^2+b^2+c^2+ab+bc+ca=2025$$$$findoutalltriplets(a,b,c).$$ Commented by Unhombre last updated…

Question Number 217244 by ArshadS last updated on 07/Mar/25 $$\mathrm{Find}\mathrm{all}\mathrm{two}-\mathrm{digit}\mathrm{numbers}\mathrm{such}\mathrm{that}$$$$\mathrm{when}\mathrm{the}\mathrm{number}\mathrm{is}\mathrm{divided}\mathrm{by}$$$$\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{its}\mathrm{digits}\mathrm{the}\mathrm{quotient}$$$$\mathrm{is}4\mathrm{and}\mathrm{the}\mathrm{remainder}\mathrm{is}3.$$ Answered by Frix last updated on 07/Mar/25…

Question Number 217129 by ArshadS last updated on 02/Mar/25 $$provethatifanintegernisnotdivisibleby2or3$$$$then{n}^2\equiv 1\left(mod24\right)$$ Commented by A5T last updated on 02/Mar/25 $$\mathrm{This}\mathrm{is}\mathrm{not}\mathrm{necessarily}\mathrm{true}.$$$$\mathrm{n}\:\mathrm{could}\:\mathrm{also}\:\mathrm{be}\:\equiv\:\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17},\mathrm{19},\mathrm{23}\:\left(\mathrm{mod}\:\mathrm{24}\right)…

Question Number 217071 by ArshadS last updated on 28/Feb/25 $$\mathrm{Find}\mathrm{all}\mathrm{two}-\mathrm{digit}\mathrm{numbers}\mathrm{that}\mathrm{are}\mathrm{equal}\mathrm{to}\mathrm{four}\mathrm{times}\mathrm{the}\mathrm{sum}$$$$\mathrm{of}\mathrm{their}\mathrm{digits}.\mathrm{Solve}\mathrm{this}\mathrm{using}\mathrm{at}\mathrm{least}\mathrm{two}\mathrm{different}\mathrm{methods}$$$$\mathrm{and}\mathrm{verify}\mathrm{your}\mathrm{answers}.$$ Answered by som(math1967) last updated on 28/Feb/25 $$\:{x}+\mathrm{10}{y}=\mathrm{4}\left({x}+{y}\right) \…