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Category: Arithmetic

Question-225599

Question Number 225599 by Jubr last updated on 04/Nov/25 Commented by Frix last updated on 04/Nov/25 $$\mathrm{If}\:{a},\:{b},\:{c},\:{d}\:\in\mathbb{R}\:\mathrm{no}\:\mathrm{maximum}\:\mathrm{exists}. \\ $$$$\mathrm{Let}\:{a}={b}=−{r};\:{c}=\mathrm{1};\:{d}=\mathrm{2}{r} \\ $$$$\left(\mathrm{1}−{r}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{r}\right)=\mathrm{1}−\mathrm{3}{r}^{\mathrm{2}} +\mathrm{2}{r}^{\mathrm{3}} \\ $$$$\underset{{r}\rightarrow+\infty}…

Question-224305

Question Number 224305 by gregori last updated on 01/Sep/25 Answered by fkwow344 last updated on 01/Sep/25 $$\mathrm{Let}'\mathrm{s}\:\mathrm{set}\:\mathrm{as}\:\overset{\rightarrow} {\boldsymbol{\mathrm{v}}}_{\mathrm{1}} =\left(\mathrm{2},\mathrm{1}\right)^{\intercal} \:,\:\overset{\rightarrow} {\boldsymbol{\mathrm{v}}}_{\mathrm{2}} =\left(\mathrm{1},\mathrm{0}\right)^{\intercal} \\ $$$${A}={PJP}^{−\mathrm{1}} \:\left(\mathrm{Jordan}\:\mathrm{decomposition}\right)…

Calculate-I-0-1-t-1-sh-t-2-dt-

Question Number 224100 by Jgrads last updated on 19/Aug/25 $$\mathrm{Calculate}\:\mathrm{I}=\underset{\:\mathrm{0}} {\int}^{\:+\infty} \left[\frac{\mathrm{1}}{\mathrm{t}}−\frac{\mathrm{1}}{\mathrm{sh}\left(\mathrm{t}\right)}\right]^{\:\mathrm{2}} \mathrm{dt} \\ $$ Answered by MathematicalUser2357 last updated on 28/Aug/25 $$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{{t}}−\frac{\mathrm{1}}{\mathrm{sinh}\:{t}}\right)^{\mathrm{2}}…

is-it-possible-to-prove-that-mn-m-n-m-n-divisible-by-6-always-

Question Number 223414 by BaliramKumar last updated on 24/Jul/25 $${is}\:{it}\:{possible}\:{to}\:{prove}\:{that}\:{mn}\left({m}+{n}\right)\left({m}−{n}\right)\: \\ $$$${divisible}\:{by}\:\mathrm{6}\:{always}\:\:\:\:\:\:\:\:\:\: \\ $$ Answered by mehdee7396 last updated on 24/Jul/25 $${let}\:\:\:{d}={mn}\left({m}+{n}\right)\left({m}−{n}\right) \\ $$$${case}\mathrm{1} \\…

Determine-gcd-13a-19b-ab-given-that-gcd-a-19-gcd-b-13-1-

Question Number 223348 by cryptograph last updated on 21/Jul/25 $${Determine}\:{gcd}\left(\mathrm{13}{a}+\mathrm{19}{b},{ab}\right)\:{given}\:{that}\:{gcd}\left({a},\mathrm{19}\right)={gcd}\left({b},\mathrm{13}\right)=\mathrm{1} \\ $$ Commented by A5T last updated on 23/Jul/25 $$\mathrm{This}\:\mathrm{isn}'\mathrm{t}\:\mathrm{unique}.\:\mathrm{a}=\mathrm{1}\:\mathrm{and}\:\mathrm{b}=\mathrm{1}\Rightarrow\:\mathrm{gcd}\left(\mathrm{32},\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{a}=\mathrm{2}\:\mathrm{and}\:\mathrm{b}=\mathrm{2}\:\Rightarrow\:\mathrm{gcd}\left(\mathrm{64},\mathrm{4}\right)=\mathrm{4} \\ $$ Answered…