Question Number 515 by 112358 last updated on 25/Jan/15 $${Find}\:{the}\:{smallest}\:{number}\:{greater} \\ $$$${than}\:{zero}\:{which}\:{can}\:{be}\:{written} \\ $$$${with}\:{ones}\:{and}\:{zeroes}\:{and}\:{is}\:{evenly}\:{divisble} \\ $$$${by}\:\mathrm{225}. \\ $$ Answered by prakash jain last updated on…
Question Number 514 by 112358 last updated on 25/Jan/15 $${Determine}\:{the}\:{smallest}\:{value}\:{of} \\ $$$${the}\:{form} \\ $$$${f}\left({u},{v}\right)=\frac{\mathrm{5}{v}^{\mathrm{2}} +\mathrm{5}{u}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{u}+{v}} \\ $$$${where}\:{u},{v}\in{R}^{+} . \\ $$ Answered by prakash jain…
Question Number 513 by 112358 last updated on 25/Jan/15 $${What}\:{is}\:{the}\:{greatest}\:{common} \\ $$$${divisor}\:{of}\:{the}\:\mathrm{2010}\:{digit}\:{and}\:\mathrm{2005}\:{digit} \\ $$$${numbers}\:{below}? \\ $$$$\mathrm{222}…\mathrm{222}\:\left(\mathrm{2010}\:{of}\:{twos}\right) \\ $$$$\mathrm{777}…\mathrm{777}\:\left(\mathrm{2005}\:{of}\:{sevens}\right) \\ $$ Answered by prakash jain last…
Question Number 503 by 123456 last updated on 20/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}−{example}: \\ $$$${if}\:{n}\in\mathbb{N},{n}>\mathrm{1},\:{exist}\:{a}\:{number}\:{k}\in\mathbb{N} \\ $$$${k}\in\left(\mathrm{0},{n}\right]\:{such}\:{that}\:{n}+{k}\:{is}\:{prime}. \\ $$ Commented by prakash jain last updated on 20/Jan/15 $$\mathrm{Bertrand}'\mathrm{s}\:\mathrm{theorem}\:\mathrm{states}\:\mathrm{that}\:…
Question Number 475 by 123456 last updated on 25/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{p}\:{is}\:{prime}\:{and}\:{n}\in\mathbb{N},\mathrm{1}<{n}\leqslant{p} \\ $$$${then}\:{pn}−\mathrm{1}\:{is}\:{prime} \\ $$ Answered by prakash jain last updated on 11/Jan/15 $${p}=\mathrm{5}…
Question Number 471 by 123456 last updated on 25/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{n}^{\mathrm{2}} \:{is}\:{prime},\:{then}\:{n}\notin\mathbb{Z} \\ $$ Answered by prakash jain last updated on 10/Jan/15 $${n}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{prime}\:\mathrm{so}\:{n}^{\mathrm{2}}…
Question Number 469 by 123456 last updated on 25/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}−{example} \\ $$$${if}\:{p}\:{is}\:{prime}\:{and}\:{a}\in\mathbb{N}\:{then} \\ $$$${p}\mid\left({a}+{p}\right)^{{p}} −{a}^{{p}} \\ $$ Answered by prakash jain last updated on 10/Jan/15…
Question Number 443 by prakash jain last updated on 04/Jan/15 $$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}\:\mathrm{minimum}\:\mathrm{value} \\ $$$$\mathrm{of}\:{n}\:\mathrm{which}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{10}^{{n}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{7}^{{p}} \right)\:\mathrm{is}\:{n}=\mathrm{6}×\mathrm{7}^{{p}−\mathrm{1}} . \\ $$ Commented by 123456 last updated…
Question Number 131497 by talminator2856791 last updated on 05/Feb/21 $$\: \\ $$$$\: \\ $$$$\:\mathrm{please}\:\mathrm{recommend}\:\mathrm{problem}\:\mathrm{and}\:\mathrm{exercise}\:\mathrm{book}\:\mathrm{for}\:\mathrm{number}\:\mathrm{theory}\: \\ $$$$\:\mathrm{where}\:\mathrm{answers}\:\mathrm{and}\:\mathrm{solutions}\:\mathrm{are}\:\mathrm{not}\:\mathrm{given}\:\mathrm{or}\:\mathrm{only} \\ $$$$\:\mathrm{given}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{book}.\:\mathrm{i}\:\mathrm{find}\:\mathrm{it}\:\mathrm{annoying}\:\mathrm{when}\:\mathrm{answers}\:\mathrm{are}\:\mathrm{always}\:\mathrm{right}\:\mathrm{next}\:\mathrm{to}\:\mathrm{the}\:\mathrm{question}. \\ $$$$\:\mathrm{a}\:\mathrm{book}\:\mathrm{that}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solutions}\:\mathrm{would}\:\mathrm{be}\:\mathrm{even}\:\mathrm{greater}. \\ $$$$\:\mathrm{the}\:\mathrm{only}\:\mathrm{such}\:\mathrm{book}\:\mathrm{i}\:\mathrm{have}\:\mathrm{found}\:\mathrm{is}\:\mathrm{250}\:\mathrm{problems}\:\mathrm{in}\:\mathrm{elementary}\:\mathrm{number}\:\mathrm{theory}. \\ $$$$\:\mathrm{thank}. \\…
Question Number 427 by 9999 last updated on 25/Jan/15 $$\mathrm{How}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{present}\:\mathrm{in} \\ $$$$\mathrm{periodic}\:\mathrm{part}\:\mathrm{for}\:\mathrm{decimal}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{7}^{\mathrm{11}} }? \\ $$ Commented by 123456 last updated on 02/Jan/15 $$\mathrm{10}^{{n}}…