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Category: Number Theory

What-is-the-greatest-common-divisor-of-the-2010-digit-and-2005-digit-numbers-below-222-222-2010-of-twos-777-777-2005-of-sevens-

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…

proof-or-given-a-counter-example-if-n-N-n-gt-1-exist-a-number-k-N-k-0-n-such-that-n-k-is-prime-

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}\:…

Prove-or-disprove-that-minimum-value-of-n-which-satisfies-the-equation-10-n-1-mod-7-p-is-n-6-7-p-1-

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…

please-recommend-problem-and-exercise-book-for-number-theory-where-answers-and-solutions-are-not-given-or-only-given-at-the-end-of-the-book-i-find-it-annoying-when-answers-are-always-right-ne

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}. \\…

How-many-digits-are-present-in-periodic-part-for-decimal-expansion-of-1-7-11-

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}}…