Question Number 112011 by Aina Samuel Temidayo last updated on 05/Sep/20 $$\mathrm{Find}\:\mid\left\{\mathrm{n}\in\mathbb{N}\mid\mathrm{n}\leqslant\mathrm{100},\:\mathrm{4}!\mid\mathrm{2}^{\mathrm{n}} −\mathrm{n}^{\mathrm{3}} \right\}\mid. \\ $$$$\left(\mathrm{Note}\:\mathrm{that}\:\mid\mathrm{S}\mid\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{cardinality}\right. \\ $$$$\left.\mathrm{or}\:\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{a}\:\mathrm{set},\mathrm{S}\right). \\ $$ Answered by Rasheed.Sindhi last updated…
Question Number 112004 by Aina Samuel Temidayo last updated on 05/Sep/20 $$\mathrm{Suppose}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\in\mathbb{N},\:\left(\mathrm{yz}+\mathrm{x}\right)\:\mathrm{is}\:\mathrm{prime} \\ $$$$\left(\mathrm{yz}+\mathrm{x}\right)\mid\left(\mathrm{zx}+\mathrm{y}\right),\:\left(\mathrm{yz}+\mathrm{x}\right)\mid\left(\mathrm{xy}+\mathrm{z}\right). \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\: \\ $$$$\frac{\left(\mathrm{xy}+\mathrm{z}\right)\left(\mathrm{zx}+\mathrm{y}\right)}{\left(\mathrm{yz}+\mathrm{x}\right)^{\mathrm{2}} }. \\ $$ Terms of Service Privacy…
Question Number 111973 by bobhans last updated on 05/Sep/20 $$\frac{\mathrm{1}}{\mathrm{5}!!}\:+\:\frac{\mathrm{1}.\mathrm{3}}{\mathrm{7}!!}\:+\:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}}{\mathrm{9}!!}\:+\:…\:+\:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}….\mathrm{95}}{\mathrm{99}!!}\:=? \\ $$ Commented by Dwaipayan Shikari last updated on 05/Sep/20 $$\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}.\mathrm{9}.\mathrm{11}…{n}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}.\mathrm{4}.\mathrm{6}.\mathrm{8}.\mathrm{10}..{n}}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} .{n}!} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{48}}…
Question Number 111831 by Aina Samuel Temidayo last updated on 05/Sep/20 $$\mathrm{Let}\:\mathrm{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{36}\:\mathrm{all} \\ $$$$\mathrm{of}\:\mathrm{whose}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{even}\:\mathrm{and}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of} \\ $$$$\mathrm{whose}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{the}\:\mathrm{same}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:\mathrm{N}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{1000}. \\ $$ Answered by Rasheed.Sindhi last updated…
Question Number 111730 by Aina Samuel Temidayo last updated on 04/Sep/20 $$\mathrm{How}\:\mathrm{many}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\:\mathrm{satisfy}\: \\ $$$$\mathrm{79x}+\mathrm{80y}+\mathrm{81z}\:=\mathrm{2016} \\ $$ Commented by kaivan.ahmadi last updated on 05/Sep/20…
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Question Number 111704 by Aina Samuel Temidayo last updated on 04/Sep/20 $$\mathrm{Assuming}\:\mathrm{FLT},\:\mathrm{prove}\:\mathrm{Fermat}−\mathrm{Euler} \\ $$$$\mathrm{theorem}:\:\left(\mathrm{a},\mathrm{n}\right)\:=\mathrm{1},\mathrm{n}\geqslant\mathrm{2}\Rightarrow\mathrm{a}^{\emptyset\left(\mathrm{n}\right)} \equiv\mathrm{1}\left(\mathrm{mod}\right. \\ $$$$\left.\mathrm{n}\right) \\ $$ Answered by Aina Samuel Temidayo last…
Question Number 46157 by Tawa1 last updated on 21/Oct/18 $$\mathrm{Please}\:\mathrm{help}. \\ $$$$\:\:\:\:\:\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\:\:\:\:\:\:\mathrm{6x}\:+\:\mathrm{8y}\:+\:\mathrm{5z}\:=\:\mathrm{101}\:. \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{got}:\:\:\:\:\:\:\:\:\mathrm{x}\:=\:−\:\mathrm{48}\:+\:\mathrm{45m}\:+\:\mathrm{4n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{y}\:=\:\:\:\:\:\mathrm{48}\:+\:\mathrm{45m}\:−\:\mathrm{3n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\:\:\:\:\:\:\mathrm{1}\:−\:\mathrm{2m} \\ $$$$ \\ $$$$\mathrm{Please}\:\mathrm{help}.\:\:\mathrm{Am}\:\mathrm{confused}.\: \\…
Question Number 111541 by Aina Samuel Temidayo last updated on 04/Sep/20 $$\mathrm{How}\:\mathrm{many}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{less}\:\mathrm{than} \\ $$$$\mathrm{1000}\:\mathrm{have}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}\:\mathrm{equal} \\ $$$$\mathrm{to}\:\mathrm{5}? \\ $$ Answered by nimnim last updated on 04/Sep/20…
Question Number 111537 by Aina Samuel Temidayo last updated on 04/Sep/20 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{obtained} \\ $$$$\mathrm{when}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{number}\:\mathrm{of}\:\mathrm{three} \\ $$$$\mathrm{different}\:\mathrm{non}−\mathrm{zero}\:\mathrm{digits}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}? \\ $$$$ \\ $$ Answered by 1549442205PVT…