Question Number 217958 by dscm last updated on 23/Mar/25 $${a},{b},{c}\in\mathbb{Z}^{+} {and} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{ab}+{bc}+{ca}=\mathrm{2025} \\ $$$${find}\:{out}\:{all}\:{triplets}\:\left({a},{b},{c}\right). \\ $$ Commented by Unhombre last updated…
Question Number 217952 by dscm last updated on 23/Mar/25 $${If}\:{x}\in\mathbb{Z}\:\wedge{y}\:{non}-{negative}\:{integer} \\ $$$${such}\:{that} \\ $$$${x}^{\mathrm{2}} +\mathrm{10}{x}+\mathrm{23}=\mathrm{2}^{{y}} \\ $$$${find}\:{out}\:{x},{y} \\ $$ Answered by Frix last updated on…
Question Number 217244 by ArshadS last updated on 07/Mar/25 $$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\: \\ $$$$\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{3}. \\ $$ Answered by Frix last updated on 07/Mar/25…
Question Number 217132 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:\:\mathrm{n}>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{\mathrm{n}−\mathrm{1}} \:+\:\mathrm{3}^{\mathrm{n}−\mathrm{1}} . \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 217129 by ArshadS last updated on 02/Mar/25 $${prove}\:{that}\:{if}\:{an}\:{integer}\:{n}\:{is}\:{not}\:{divisible}\:{by}\:\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\:{then}\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\:\mathrm{24}\right) \\ $$ Commented by A5T last updated on 02/Mar/25 $$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{necessarily}\:\mathrm{true}.\: \\ $$$$\mathrm{n}\:\mathrm{could}\:\mathrm{also}\:\mathrm{be}\:\equiv\:\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17},\mathrm{19},\mathrm{23}\:\left(\mathrm{mod}\:\mathrm{24}\right)…
Question Number 217130 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{integer}\:\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{n}^{\mathrm{4}} +\:\mathrm{4}^{{n}} \:\:\mathrm{is} \\ $$$$\mathrm{c}{o}\mathrm{mposite}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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 $$\:{x}+\mathrm{10}{y}=\mathrm{4}\left({x}+{y}\right) \\…
Question Number 217066 by ArshadS last updated on 28/Feb/25 $${Find}\:{all}\:{integer}\:{x},{y}\:{such}\:{that} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{100} \\ $$ Answered by mehdee7396 last updated on 28/Feb/25 $$\left({x}−{y}\right)\left({x}+{y}\right)=\mathrm{2}×\mathrm{50}=\mathrm{10}×\mathrm{10} \\…
Question Number 217030 by ArshadS last updated on 27/Feb/25 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that}\:\: \\ $$$$\:\mathrm{n}\:+\:\mathrm{1}\:\:\mathrm{divides}\:\:\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$ Answered by issac last updated on 27/Feb/25 $$\frac{{n}+\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\frac{{n}+\mathrm{1}}{\left(\mathrm{1}+{n}\boldsymbol{{i}}\right)\left(\mathrm{1}−{n}\boldsymbol{{i}}\right)} \\…
Question Number 217040 by ArshadS last updated on 27/Feb/25 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{{n}} \:+\:\mathrm{1}.\:\: \\ $$ Answered by Ghisom last updated on 27/Feb/25 $$\mathrm{one}\:\mathrm{group}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{is}\:{n}=\mathrm{3}^{{k}} \wedge{k}\in\mathbb{N} \\…