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Category: Number Theory

Let-S-1-2-3-4-48-49-What-is-the-maximum-value-of-n-such-that-it-is-possible-to-select-n-numbers-from-S-and-arrange-them-in-a-circle-in-such-a-way-that-the-product-of-any-two-adjacent-numbers

Question Number 116356 by bemath last updated on 03/Oct/20 $$\mathrm{Let}\:\mathrm{S}\:=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},…,\mathrm{48},\mathrm{49}\right\}\:.\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{select}\:\mathrm{n}\:\mathrm{numbers}\: \\ $$$$\mathrm{from}\:\mathrm{S}\:\mathrm{and}\:\mathrm{arrange}\:\mathrm{them}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\: \\ $$$$\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{adjacent}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{100}? \\ $$ Terms…

what-the-cooefficient-of-x-10-from-1-x-1-2x-2-1-3x-3-1-2020x-2020-

Question Number 115742 by bemath last updated on 28/Sep/20 $${what}\:{the}\:{cooefficient}\:{of}\:{x}^{\mathrm{10}} \:{from} \\ $$$$\left(\mathrm{1}+{x}\right)×\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)×\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right)×…×\left(\mathrm{1}+\mathrm{2020}{x}^{\mathrm{2020}} \right) \\ $$ Answered by mr W last updated on…

Q-find-the-largest-value-of-such-that-the-positive-integers-a-b-gt-1-satisfy-a-b-b-a-a-b-b-a-5329-

Question Number 180877 by Sheshdevsahu last updated on 18/Nov/22 $$\boldsymbol{\mathrm{Q}}.\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{largest}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}}\:\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}}\:>\:\mathrm{1}\:\boldsymbol{\mathrm{satisfy}}. \\ $$$$\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{b}}} .\boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{a}}} \:+\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{b}}} \:+\:\boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{a}}} \:=\:\mathrm{5329} \\ $$ Commented by MJS_new last updated…

Simplify-4-2-3-

Question Number 180776 by Acem last updated on 17/Nov/22 $${Simplify}\:\:\sqrt{\frac{\mathrm{4}}{\:\sqrt{\mathrm{2}}}\:+\:\mathrm{3}}\: \\ $$$$ \\ $$ Answered by BaliramKumar last updated on 17/Nov/22 $$\sqrt{\frac{\mathrm{4}}{\:\sqrt{\mathrm{2}}}\:+\:\mathrm{3}\:} \\ $$$$\sqrt{\mathrm{2}\sqrt{\mathrm{2}\:}\:+\:\mathrm{3}\:} \\…

Question-180746

Question Number 180746 by Acem last updated on 16/Nov/22 Answered by MJS_new last updated on 16/Nov/22 $$\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{200}}\right)^{\mathrm{200}} >\mathrm{2} \\ $$$$\mathrm{because}\:\mathrm{we}\:\mathrm{know} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}}\right)^{\mathrm{1}} =\mathrm{2} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}}…

Question-180115

Question Number 180115 by universe last updated on 07/Nov/22 Answered by mr W last updated on 07/Nov/22 $$\mathrm{7}×\mathrm{6}×\mathrm{4}×\mathrm{5}×\mathrm{2}=\mathrm{1680} \\ $$$$=\mathrm{40}×\mathrm{42}=\left(\mathrm{42}−\mathrm{1}\right)\left(\mathrm{41}+\mathrm{1}\right)=\mathrm{41}^{\mathrm{2}} −\mathrm{1} \\ $$$$\Rightarrow\:\left({d}\right) \\ $$…

find-four-numbers-such-that-the-sum-of-every-two-and-the-sum-of-all-four-should-be-perfect-squaresk-

Question Number 48616 by amankumar last updated on 26/Nov/18 $${find}\:{four}\:{numbers}\:{such}\:{that}\:{the}\:{sum}\:{of}\:{every}\:{two}\:{and}\:{the}\:{sum}\:{of}\:{all}\:{four}\:{should}\:{be}\:{perfect}\:{squaresk}\: \\ $$ Commented by MJS last updated on 26/Nov/18 $${a}\neq{b}\neq{c}\neq{d}? \\ $$$${a},{b},{c},{d}\:\in\mathbb{N}\:\mathrm{or}\:\in\mathbb{Z}?\:\neq\mathrm{0}? \\ $$$$\mathrm{please}\:\mathrm{make}\:\mathrm{it}\:\mathrm{clear} \\…