Category: Number Theory

Question Number 18652 by Tinkutara last updated on 26/Jul/17 $$\mathrm{Show}\mathrm{that}\mathrm{for}\mathrm{any}\mathrm{natural}\mathrm{number}n,$$$$\mathrm{the}\mathrm{fraction}\frac{21n+4}{14n+3}\mathrm{is}\mathrm{in}\mathrm{its}\mathrm{lowest}\mathrm{term}.$$ Answered by diofanto last updated on 27/Jul/17 $$\mathrm{Lets}\mathrm{prove}\mathrm{by}\mathrm{contradiction}.$$$$\mathrm{Suppose}\:\mathrm{there}\:\mathrm{is}\:{d}\:>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \…

Question Number 83891 by bagjamath last updated on 07/Mar/20 $$$$$$$$$${1111}^{2019}\mathrm{mod}11111=\dots .?$$ Commented by MJS last updated on 08/Mar/20 $$\mathrm{you}\:\mathrm{are}\:\mathrm{right},\:\mathrm{I}\:\mathrm{misread}\:\mathrm{1111}^{\mathrm{2019}}…

Question Number 148951 by EDWIN88 last updated on 01/Aug/21 $$Letcomplexnumberz=(a+\cos \theta )+(2a-\sin \theta )i.$$$$If\mid z\mid ⩽2forany\theta \in Rthenthe$$$$rangeofrealnumberais\mathrm{\_}\mathrm{\_}\mathrm{\_}$$ Answered by iloveisrael last updated on 01/Aug/21 Answered by…

Question Number 17446 by Tinkutara last updated on 05/Jul/17 $$\mathrm{Find}\mathrm{the}\mathrm{integer}\mathrm{closest}\mathrm{to}$$$$100(12-\sqrt{143}).$$ Commented by ajfour last updated on 06/Jul/17 $$100(12-\sqrt{143})=\frac{100}{12+\sqrt{143}}$$$$\:\:\approx\frac{\mathrm{100}}{\mathrm{24}}\:\approx\:\mathrm{4}.\mathrm{2}\: \…

Question Number 17272 by RasheedSoomro last updated on 03/Jul/17 $$\mathrm{Determine}\mathrm{two}\mathrm{distinct}\mathrm{primes}\mathrm{p}\mathrm{and}\mathrm{q}$$$$\mathrm{such}\mathrm{that}:$$$$\left(\mathrm{i}\right)\mathrm{p}+\mathrm{q}+1,\mathrm{p}+\mathrm{q}-1,\frac{\mathrm{p}+\mathrm{q}}{2}\in \mathbb{P}\left(\mathrm{All}\mathrm{primes}\right)?$$$$\left(\mathrm{ii}\right)\mathrm{p}+\mathrm{q}+1,\mathrm{p}+\mathrm{q}-1,\frac{\mathrm{p}+\mathrm{q}}{2},\frac{\mathrm{p}-\mathrm{q}}{2}\in \mathbb{P}\left(\mathrm{All}\mathrm{primes}\right)?$$ Commented by prakash jain last updated on…