Category: Number Theory

Question Number 110562 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Let}\mathrm{x}\mathrm{and}\mathrm{y}\mathrm{be}\mathrm{integers}\mathrm{such}\mathrm{that}$$$$\mathrm{xy}\ne 1,{\mathrm{x}}^2\ne \mathrm{y}\mathrm{and}{\mathrm{y}}^2\ne \mathrm{x}.$$$$$$$$\left(\mathrm{i}\right)\mathrm{Show}\mathrm{that}\mathrm{p}\mid \mathrm{xy}-1\mathrm{and}\mathrm{p}\mid {\mathrm{x}}^2-\mathrm{y}$$$$\mathrm{then}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x}\:\mathrm{where}\:\mathrm{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}. \…

Question Number 44986 by Tawa1 last updated on 07/Oct/18 $$\mathrm{Find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{nth}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{series}:\frac{1}{1.2.3}+\frac{1}{4.5.6}+\frac{1}{7.8.9}+\dots$$ Commented by tanmay.chaudhury50@gmail.com last updated on 07/Oct/18 $$plswritetheanswerofquestiinssndsourceof$$$$question\dots$$ Answered…

Question Number 110358 by bobhans last updated on 28/Aug/20 $$ifpositiveintegerxsatisfies{x}^2-4x+56\equiv 14\left(mod17\right)$$$$,whatistheminimumvalueofx.$$ Answered by john santu last updated on 28/Aug/20 $$\Leftrightarrow{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}\:+\:\mathrm{52}\:=\:\mathrm{14}\:\left({mod}\:\mathrm{17}\right)…

Question Number 110354 by Aina Samuel Temidayo last updated on 28/Aug/20 $$\mathrm{The}\mathrm{Diophantine}\mathrm{equation}$$$${\mathrm{x}}^2+{\mathrm{y}}^2+1=\mathrm{N}(\mathrm{xy}+1)\mathrm{has}$$$$\mathrm{infinitely}\mathrm{many}\mathrm{integer}$$$$\mathrm{solutions}\mathrm{if}\mathrm{N}\mathrm{equals}?$$ Commented by Aina…

Question Number 44716 by Tawa1 last updated on 03/Oct/18 $$\mathrm{prove}:1+11+111+\dots .+\frac{111\dots 111}{\mathrm{n}\mathrm{times}}=\frac{{10}^{\mathrm{n}+1}-9\mathrm{n}-10}{81}$$ Answered by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18 $$s=1+11+111+1111+\dots +111..\underset{ntimes}{.}111$$$$\mathrm{9}{s}=\mathrm{9}+\mathrm{99}+\mathrm{999}+\mathrm{9999}+…+\mathrm{999}…\mathrm{999} \…