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Category: Number Theory

a-b-c-N-such-that-a-3-b-b-3-c-Q-show-that-a-2-b-2-c-2-a-b-c-Z-

Question Number 113187 by bobhans last updated on 11/Sep/20 $$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\mathbb{N}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{a}\sqrt{\mathrm{3}}\:+\mathrm{b}}{\mathrm{b}\sqrt{\mathrm{3}}+\mathrm{c}}\:\in\:\mathrm{Q},\:\mathrm{show} \\ $$$$\mathrm{that}\:\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}+\mathrm{c}}\:\in\:\mathbb{Z} \\ $$ Answered by bemath last updated on 11/Sep/20 $$\mathrm{Fact}\::\:\sqrt{\mathrm{3}}\:\notin\mathrm{Q}\:,\:\mathrm{p}+\mathrm{q}\sqrt{\mathrm{3}}\:\in\mathrm{Q}\:\mathrm{if}\:\mathrm{q}=\mathrm{0}…

any-one-can-explain-me-how-to-change-decimal-number-to-biner-number-i-m-forgot-example-315-10-2-thank-you-

Question Number 113134 by bemath last updated on 11/Sep/20 $$\mathrm{any}\:\mathrm{one}\:\mathrm{can}\:\mathrm{explain}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to} \\ $$$$\mathrm{change}\:\mathrm{decimal}\:\mathrm{number}\:\mathrm{to}\: \\ $$$$\mathrm{biner}\:\mathrm{number}.\:\mathrm{i}'\mathrm{m}\:\mathrm{forgot}. \\ $$$$\mathrm{example}\:\left(\mathrm{315}\right)_{\mathrm{10}} \:=\:\left(…\right)_{\mathrm{2}} \\ $$$$\mathrm{thank}\:\mathrm{you} \\ $$ Commented by I want…

Prove-that-GCD-a-b-b-a-b-

Question Number 113101 by deArchie last updated on 11/Sep/20 $${Prove}\:{that}\:{GCD}\:\left(\left({a},{b}\right),{b}\right)=\left({a},{b}\right) \\ $$ Commented by MJS_new last updated on 11/Sep/20 $$\mathrm{how}\:\mathrm{do}\:\mathrm{you}\:\mathrm{define}\:\mathrm{the}\:\mathrm{gcd}\:\mathrm{of}\:\mathrm{a}\:\mathrm{number}\:\mathrm{with} \\ $$$$\mathrm{a}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{numbers}?\:\mathrm{check}\:\mathrm{the}\:\mathrm{syntax} \\ $$ Commented…

What-is-the-area-bounded-by-the-curves-arg-z-pi-3-arg-z-2pi-3-and-arg-z-2-2i-3-pi-on-the-complex-plane-

Question Number 113091 by bobhans last updated on 11/Sep/20 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curves} \\ $$$$\mathrm{arg}\left(\mathrm{z}\right)\:=\:\frac{\pi}{\mathrm{3}}\:;\:\mathrm{arg}\left(\mathrm{z}\right)=\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:\mathrm{and}\:\mathrm{arg}\left(\mathrm{z}−\mathrm{2}−\mathrm{2i}\sqrt{\mathrm{3}}\right)=\pi \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}? \\ $$ Answered by john santu last updated on 11/Sep/20 $$\:\left({i}\right)\:{arg}\:\left({z}\right)\:=\:\frac{\pi}{\mathrm{3}}\:\rightarrow\:\frac{{y}}{{x}}\:=\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{3}}\right)…

Find-all-positive-integers-n-for-which-5-n-1-is-divisible-by-7-

Question Number 113073 by bobhans last updated on 11/Sep/20 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}\:\mathrm{for}\: \\ $$$$\mathrm{which}\:\mathrm{5}^{\mathrm{n}} +\mathrm{1}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7} \\ $$ Answered by john santu last updated on 11/Sep/20 $$\:\:{find}\:{all}\:{positive}\:{integers}\:{n}\:{such} \\…

a-1-b-1-c-1-d-2-1-3-If-a-b-c-d-are-positive-integers-then-what-is-the-value-of-b-

Question Number 113006 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{a}+\frac{\mathrm{1}}{\mathrm{b}+\frac{\mathrm{1}}{\mathrm{c}+\frac{\mathrm{1}}{\mathrm{d}+…}}}=\sqrt[{\mathrm{3}}]{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},…\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:'\mathrm{b}'? \\ $$ Answered by mr W last updated on…

After-distributing-sweets-equally-among-25-children-8-sweets-remained-Had-the-number-of-children-been-28-22-sweets-would-have-been-left-after-equally-distributing-What-was-the-total-number-of-swee

Question Number 113002 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{After}\:\mathrm{distributing}\:\mathrm{sweets}\:\mathrm{equally} \\ $$$$\mathrm{among}\:\mathrm{25}\:\mathrm{children},\:\mathrm{8}\:\mathrm{sweets} \\ $$$$\mathrm{remained}.\:\mathrm{Had}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{children} \\ $$$$\mathrm{been}\:\mathrm{28},\:\mathrm{22}\:\mathrm{sweets}\:\mathrm{would}\:\mathrm{have}\:\mathrm{been} \\ $$$$\mathrm{left}\:\mathrm{after}\:\mathrm{equally}\:\mathrm{distributing}.\:\mathrm{What} \\ $$$$\mathrm{was}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{sweets}? \\ $$ Answered…

Two-different-two-digit-natural-numbers-are-written-beside-each-other-such-that-the-larger-number-is-written-on-the-left-When-the-absolute-difference-of-the-two-numbers-is-subtracted-from-the-four-di

Question Number 113005 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{Two}\:\mathrm{different}\:\mathrm{two}−\mathrm{digit}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{written}\:\mathrm{beside}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{larger}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{written}\:\mathrm{on}\:\mathrm{the}\:\mathrm{left}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{absolute}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{is}\:\mathrm{subtracted}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{number}\:\mathrm{so}\:\mathrm{formed},\:\mathrm{the} \\…

The-first-23-natural-numbers-are-written-in-an-increasing-order-beside-each-other-to-form-a-single-number-What-is-the-remainder-when-this-number-is-divided-by-18-

Question Number 113000 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{The}\:\mathrm{first}\:\mathrm{23}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{order} \\ $$$$\mathrm{beside}\:\mathrm{each}\:\mathrm{other}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{number}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{this}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{18}? \\ $$ Answered by floor(10²Eta[1])…