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Category: Number Theory

How-many-natural-numbers-less-than-1000-have-the-sum-of-their-digits-equal-to-5-

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…

What-is-the-minimum-value-obtained-when-an-arbitrary-number-of-three-different-non-zero-digits-is-divided-by-the-sum-of-its-digits-

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…

Find-four-values-of-n-satisfying-1-n-2000-and-2-n-n-2-mod-1024-

Question Number 111503 by Aina Samuel Temidayo last updated on 04/Sep/20 $$\mathrm{Find}\:\mathrm{four}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\:\mathrm{satisfying} \\ $$$$\mathrm{1}\leqslant\mathrm{n}\leqslant\mathrm{2000}\:\mathrm{and}\:\mathrm{2}^{\mathrm{n}} =\mathrm{n}^{\mathrm{2}} \left(\mathrm{mod}\:\mathrm{1024}\right) \\ $$ Answered by 1549442205PVT last updated on 04/Sep/20…

Find-the-number-of-rational-numbers-r-0-lt-r-lt-1-such-that-when-r-is-written-as-a-fraction-in-lowest-term-The-numerator-and-the-denominator-have-a-sum-of-1000-

Question Number 111155 by Aina Samuel Temidayo last updated on 02/Sep/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers} \\ $$$$\mathrm{r},\:\mathrm{0}<\mathrm{r}<\mathrm{1},\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{r}\:\mathrm{is}\:\mathrm{written} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{lowest}\:\mathrm{term}.\:\mathrm{The} \\ $$$$\mathrm{numerator}\:\mathrm{and}\:\mathrm{the}\:\mathrm{denominator} \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{1000}. \\ $$ Answered by Her_Majesty…

Show-that-the-square-of-every-odd-integer-is-of-the-form-8m-1-

Question Number 45439 by Tawa1 last updated on 12/Oct/18 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{every}\:\mathrm{odd}\:\mathrm{integer}\:\mathrm{is}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\:\:\mathrm{8m}\:+\:\mathrm{1} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 13/Oct/18 $$\left(\mathrm{2}{k}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{4}{k}^{\mathrm{2}} +\mathrm{4}{k}+\mathrm{1} \\ $$$$=\mathrm{4}{k}\left({k}+\mathrm{1}\right)+\mathrm{1} \\…

How-many-ways-can-2018-be-expressed-as-the-sum-of-two-squares-

Question Number 110895 by Aina Samuel Temidayo last updated on 31/Aug/20 $$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{2018}\:\mathrm{be} \\ $$$$\mathrm{expressed}\:\mathrm{as}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{squares}? \\ $$ Answered by ajfour last updated on 31/Aug/20 $$\mathrm{43}^{\mathrm{2}} +\mathrm{13}^{\mathrm{2}}…